The Adams differentials on the classes $h_j^3$

TL;DR

This paper establishes new non-trivial differentials on classes in the Adams spectral sequence at filtration 3, using advanced deformations of stable homotopy theory, and confirms a conjecture of Mahowald.

Contribution

It proves an infinite family of non-trivial $d_4$-differentials on classes $h_j^3$ for $j geq 6$, advancing understanding of the Adams spectral sequence.

Findings

Proves non-trivial $d_4$-differentials on $h_j^3$ for $j geq 6$

Shows $h_j^2$ survives to the Adams $E_5$-page

Shows $h_6^2$ survives to the Adams $E_9$-page

Abstract

In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes $h_j$, resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill--Hopkins--Ravenel proved that the classes $h_j^2$ support non-trivial differentials for $j \geq 7$, resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes $h_j^2$ for $j \geq 7$ and the fate of $h_6^2$ remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial $d_4$-differentials on the classes $h_j^3$ for $j \geq 6$, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory -- $\mathbb{C}$-motivic stable homotopy theory and $\mathbb{F}_2$-synthetic homotopy theory -- both in an essential way. Along the way, we also show that $h_j^2$ survives to the Adams $E_5$-page and that $h_6^2$ survives to the Adams $E_9$-page.

Equations

$$d_r(h_j^2)=0\text{for}2\le r\le 4.$$

$$d_r(h_6^2)=0\text{for}2\le r\le 8.$$

$${\mathrm{Sq}}^0:{\mathrm{Ext}}_A^{s,t}({\mathbb{F}}_2,{\mathbb{F}}_2)⟶{\mathrm{Ext}}_A^{s,2t}({\mathbb{F}}_2,{\mathbb{F}}_2).$$

$${\mathrm{Sq}}^0(h_j)=h_{j+1}.$$

$$d_r(a_j)=c\cdot b_j\ne 0$$

$${\mathrm{Ext}}_A^{s,t}(F_p,F_p)$$

$${\mathrm{Ext}}_{{\mathrm{BP}}_{\ast }\mathrm{BP}}^{s,t}({\mathrm{BP}}_{\ast },{\mathrm{BP}}_{\ast })$$

$${\mathrm{Ext}}_{{\mathrm{BP}}_{\ast }\mathrm{BP}}^{s,t}({\mathrm{BP}}_{\ast },{\mathrm{BP}}_{\ast })⟶{\mathrm{Ext}}_A^{s,t}(F_p,F_p)$$

$${\mathrm{Ext}}_{{\mathrm{BP}}_{\ast }\mathrm{BP}/I}^{s,t^{\prime }}({\mathrm{BP}}_{\ast }/I,I^k/I^{k+1})\cong E_2^{s,k,t^{\prime }}⟹{\mathrm{Ext}}_{{\mathrm{BP}}_{\ast }\mathrm{BP}}^{s,t^{\prime }}({\mathrm{BP}}_{\ast },{\mathrm{BP}}_{\ast })$$

$$d_r:E_r^{s,k,t^{\prime }}⟶E_r^{s+1,k+r-1,t^{\prime }}$$

$$\mathcal{P}\cong F_2[{\xi }_1^2,{\xi }_2^2,\cdots ]\subseteq F_2[{\xi }_1,{\xi }_2,\cdots ]\cong A$$

$$E_2^{s,k,t}={\mathrm{Ext}}_{\mathcal{P}}^{s,t}(F_p,{\mathrm{Ext}}_{\mathcal{Q}}^k(F_p,F_p))⟹{\mathrm{Ext}}_A^{s+k,t}(F_p,F_p)$$

$$d_r:E_r^{s,k,t}⟶E_r^{s+r,k-r-1,t}$$

$${\mathrm{Ext}}_{\mathcal{Q}}^{\ast ,\ast }(F_2,F_2)\cong F_2[q_0,q_1,\cdots ],$$

$$({\mathrm{BP}}_{\ast }/I,{\mathrm{BP}}_{\ast }\mathrm{BP}/I)\cong (F_p,\mathcal{P})$$

$${\mathrm{Ext}}_{\mathcal{Q}}^{\ast }(F_p,F_p)\cong F_p[q_0,q_1,\cdots ]\cong F_p[v_0,v_1,\cdots ]\cong {\oplus }_{\ast }{I}^{\ast }/I^{\ast +1}$$

$${\mathrm{Ext}}_{\mathcal{P}}^{s,t}(F_p,{\mathrm{Ext}}_{\mathcal{Q}}^k(F_p,F_p))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Ext}}_{\mathcal{P}}^{s,t}(F_p,{\mathrm{Ext}}_{\mathcal{Q}}^k(F_p,F_p))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tau :S^{0,-1}⟶S^{0,0}$$

$$S^{0,-1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbf{Re}(S^{n,w})$$

$$\mathbf{Re}(F_p^{\text{mot}})$$

$$\mathbf{Re}(\text{BPGL})$$

$$A^{\text{mot}}\cong F_2[\tau ][{\tau }_1,{\tau }_2,\cdots ][{\xi }_1^2,{\xi }_2^2,\cdots ]/{\tau }_i^2=\tau {\xi }_i^2,$$

$$\mathbf{Re}(\tau )=1,\mathbf{Re}({\tau }_i)={\xi }_i,\mathbf{Re}({\xi }_i^2)={\xi }_i^2.$$

$${\mathrm{Ext}}_{A^{\text{mot}}}^{a,t,w}(F_p[\tau ],F_p[\tau ])\cong E_2^{a,t,w}⟹{\pi }_{t-a,w}{S}^{0,0}$$

$$d_r:E_r^{a,t,w}\to E_r^{a+r,t+r-1,w}$$

$${\mathrm{Ext}}_{A^{\text{mot}}}^{a,t,w}(F_p[\tau ],F_p[\tau ])⟶{\mathrm{Ext}}_{A^{\text{mot}}}^{a,t,w}(F_p[\tau ],F_p).$$

$${\mathrm{Ext}}_{A^{\text{mot}}}^{s+k,t^{\prime }-k,\frac{t^{\prime }}{2}}(F_p[\tau ],F_p)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Ext}}_A^{a,t}(F_p,F_p)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Ext}}_{\mathcal{P}}^{s,t}(F_p,{\mathrm{Ext}}_{\mathcal{Q}}^k(F_p,F_p))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Ext}}_{{\text{BPGL}}_{\ast ,\ast }\text{BPGL}}^{s,t^{\prime },w}({\text{BPGL}}_{\ast ,\ast },{\text{BPGL}}_{\ast ,\ast })\cong E_2^{s,t^{\prime },w}⟹{\pi }_{t^{\prime }-s,w}{S}^{0,0}$$

Full text

In this paper, in Adams filtration 3, we prove an infinite family of non-trivial $d_{4}$-differentials on the classes $h_{j}^{3}$ for $j\geq 6$, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory—$\mathbb{C}$-motivic stable homotopy theory and $\mathbb{F}_{2}$-synthetic homotopy theory—both in an essential way. Along the way, we also show that $h_{j}^{2}$ survives to the Adams $E_{5}$-page and that $h_{6}^{2}$ survives to the Adams $E_{9}$-page.

1. 1 Introduction
2. 2 The Miller square and motivic homotopy theory
3. 3 Proof of the main theorem
4. 4 The Cartan–Eilenberg $E_{2}$-page
5. 5 The Cartan–Eilenberg spectral sequence
6. 6 The key algebraic Novikov differential
7. 7 The inductive approach revisited
8. A Code

1. Introduction

One of the basic goals of stable homotopy theory is to obtain a systematic understanding of the differentials in the Adams spectral sequence. Perhaps surprisingly, many questions in differential topology also require and reduce to a core Adams spectral sequence computation. As an example, let us consider the following question: When is the $n$-sphere parallelizable? Classical constructions using division algebras provide parallelizations of $S^{1}$, $S^{3}$ and $S^{7}$ and the remaining problem is to show that all other spheres are not parallelizable. For this it suffices to show that there does not exist a class with Hopf invariant one in the $n$-stem.In order to approach this problem Adams began by reformulating earlier work of Adem into the following theorem.

Theorem 1.1** (Adem, Adams [Ada58]).**

Hopf invariant one classes can only exist in dimensions of the form $2^{j}-1$. Moreover, there exists a Hopf invariant one class in the $(2^{j}-1)$-stem if and only if the class $h_{j}$ on the $1$-line of the Adams spectral sequence is a permanent cycle.

Then, using secondary cohomology operations, Adams computed the first infinite family of nonzero differentials in the Adams spectral sequence, resolving the Hopf invariant one problem.

Theorem 1.2** (Adams [Ada60]).**

$d_{2}(h_{j})=h_{0}h_{j-1}^{2}\neq 0$* for all $j\geq 4$.*

Remark 1.3**.**

We now have many other methods proving Adams’ theorem, for example, using Adams operations in K-theory, using the comparison map between the Adams and the Adams-Novikov spectral sequence, using product structure in higher Adams filtrations, and using Bruner’s power operation in the Adams spectral sequence.

On the Adams $2$-line we have classes $h_{j}^{2}$ and Browder showed that they too are intimately connected with a fundamental problem in differential topology.

Theorem 1.4** (Browder [Bro69]).**

A smooth framed manifold with Kervaire invariant one can only exist in dimensions of the form $2(2^{j}-1)$. Moreover, the following statements are equivalent.

• •

The class $h_{j}^{2}$ is a permanent cycle in the Adams spectral sequence.

• •

There exists a smooth framed manifold of dimension $2(2^{j}-1)$ with Kervaire invariant one.

Using equivariant and chromatic methods, Hill, Hopkins and Ravenel proved the following celebrated theorem, resolving the Kervaire invariant problem in large dimensions.

Theorem 1.5** (Hill–Hopkins–Ravenel [HHR16]).**

The classes $h_{j}^{2}$ support nonzero Adams differentials for $j\geq 7$.

Remark 1.6**.**

Unlike the solution of the Hopf invariant one problem, the Hill–Hopkins–Ravenel proof is still the only proof to the solution of the Kervaire invariant one problem. Also note that the targets of the Adams differentials on the Kervaire classes $h_{j}^{2}$ have not been identified.

Computations of the homotopy groups of spheres show that the Kervaire classes $h_{j}^{2}$ are permanent cycles for $0\leq j\leq 5$ [BMT70, BJM84, Xu16]. The final case, the fate of $h_{6}^{2}$ in dimension 126, remains open.

Continuing in this manner we are led to consider the classes $h_{j}^{3}$ on the Adams $3$-line. We begin by asking whether they too lie at the heart of some geometric problem.

Question 1.7**.**

Do the classes $h_{j}^{3}$ also have an interpretation in differential topology?

Barratt, Mahowald and Tangora [BMT70] proved that the classes $h_{j}^{3}$ are permanent cycles for $j\leq 4$. At Toda’s $60^{\mathrm{th}}$ birthday conference in 1988 Mahowald then made the following conjecture (see [Min95, Remark 3.1]) regarding the $h_{j}^{3}$ family:

Conjecture 1.8** (Mahowald).**

The classes $h_{j}^{3}$ are not permanent cycles for $j$ large.

Recently, Isaksen, Wang and the second author proved that $h_{5}^{3}$ supports a nonzero Adams differential using motivic stable homotopy theory. Let $g_{j}$ be the generator of the group $\operatorname{\mathrm{Ext}}^{4,3\cdot 2^{j+2}}_{\operatorname{\mathcal{A}}}(\mathbb{F}_{2},\mathbb{F}_{2})\cong\mathbb{F}_{2}$.

Theorem 1.9** (Isaksen–Wang–Xu [IWX20b]).**

$d_{4}(h_{5}^{3})=h_{0}^{3}g_{3}\neq 0$.

Previously, Wu and Lin had, respectively, shown that $h_{6}^{3}$ and $h_{7}^{3}$ support nonzero Adams differentials.

Theorem 1.10** (Lin [Lin98], Wu [Wu13]).**

$d_{4}(h_{6}^{3})=h_{0}^{3}g_{4}\neq 0$* and $d_{4}(h_{7}^{3})\neq 0$.*

Lin and Wu analyze $h_{6}^{3}$ and $h_{7}^{3}$ via a study of the Kahn–Priddy transfer map. Lin also went on to conjecture that $d_{4}(h_{j}^{3})=h_{0}^{3}g_{j-2}$ for $j\geq 6$ [Lin98] and Wu verified this conjecture in the case $j=6$.

In this article we prove Lin’s refinement of Mahowald’s Conjecture, thereby determining the fate the $h_{j}^{3}$ family.

Theorem 1.11**.**

$d_{4}(h_{j}^{3})=h_{0}^{3}g_{j-2}\neq 0$* for $j\geq 6$.*

Remark 1.12**.**

We would like to comment that we don’t expect the fate of $h_{j}^{3}$ can be obtained via the Hill–Hopkins–Ravenel’s detection method for cyclic groups $C_{2^{n}}$. This is partly because we will show that for $j$ large the class $h_{j}^{3}$ does not survive to the Adams–Novikov $E_{2}$-page in the algebraic Novikov spectral sequence so it is difficult to directly compare with the homotopy fixed point spectral sequence and the slice spectral sequence; This is also suggested by a theorem of Hill [Hil15] that $\eta^{3}$ maps to zero under the Hurewicz map for all fixed points of all norms to cyclic 2-groups of the Landweber-Araki Real bordism spectrum, and the work of Li–Shi–Wang–Xu [LSWX19] that the Hurewicz image of the $C_{2}$-fixed point of the Real bordism spectrum behaves well under the $\operatorname{\mathrm{Sq}}^{0}$-operation (see also 1.17).

The proof of 1.11 uses two different deformations of the category of spectra—$\mathbb{C}$-motivic spectra and $\operatorname{\mathbb{F}}_{2}$-synthetic spectra—in an essential way. We prove 1.11 as the Betti realization of a corresponding differential in the motivic Adams spectral sequence. This differential in turn is lifted from the the motivic Adams spectral sequence of a certain 2 cell complex—the cofiber of $\tau$. By a theorem of Gheorghe, Wang and the second author [GWX21, Theorem 1.3], the motivic Adams spectral sequence for the cofiber of $\tau$ is isomorphic to the algebraic Novikov spectral sequence. The crucial step in our proof is then to establish a certain non-trivial differential in the algebraic Novikov spectral sequence.

The most delicate part of our argument is in lifting the algebraic Novikov differential proved in Section 6 to the motivic Adams spectral sequence and it is at this point where we need several auxiliary inputs from $\operatorname{\mathbb{F}}_{2}$-synthetic spectra. In Section 7 we analyze the differentials on the classes $h_{j}^{2}$ using a synthetic refinement of the “inductive approach to Kervaire invariant one” from [BJM83]. As a consequence of this we obtain the following theorem (see Section 7 for our conventions regarding synthetic spectra.).

Theorem 1.13**.**

*Fix an $r\geq 2$ and suppose that $\theta_{j}$ is a lift of $h_{j}^{2}$ from $\operatorname{\mathbb{S}}/\!\lambda$ to $\operatorname{\mathbb{S}}/\!\lambda^{r}$. If $2\theta_{j}=0$ and $\lambda^{2}\theta_{j}^{2}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{r})$, then there exists a class $\theta_{j+1}$ lifting $h_{j+1}^{2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{r}$ such that $2\theta_{j+1}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{r})$. *

As a corollary of 1.13 we show that

Corollary 1.14**.**

The class $h_{j}^{2}$ survives to the Adams $E_{5}$-page for all $j\geq 0$. In other words,

[TABLE]

Specializing to the case $j=6$ we are able to refine our arguments to obtain partial progress towards understanding the fate of $h_{6}^{2}$.

Theorem 1.15**.**

The class $h_{6}^{2}$ survives to the Adams $E_{9}$-page. In other words,

[TABLE]

Remark 1.16**.**

Adams proved that $h_{j}^{4}=0$ for all $j\geq 1$ and $h_{0}^{n}$ is a nonzero permanent cycles detecting $2^{n}$, therefore as a consequence of 1.11 the only remaining class of the form $h_{j}^{n}$ whose fate we do not know is $h_{6}^{2}$.

The New Doomsday Conjecture

The infinite families of Adams differentials in Theorems 1.2 and 1.11 share a key structural feature: their sources and targets naturally fit into $\operatorname{\mathrm{Sq}}^{0}$-families. We expect that this is not a coincidence.

Recollection 1.17**.**

The commutative $\operatorname{\mathbb{F}}_{2}$-algebra structure on the stack of additive formal groups provides us with algebraic Steenrod operations (see Chapters IV–VI of [BMMS86]) acting on the cohomology of the Steenrod algebra. In particular, there is an operation

[TABLE]

Given a class $x$ in the cohomology of the Steenrod algebra we obtain an infinite $\operatorname{\mathrm{Sq}}^{0}$-family of classes $\{x,\operatorname{\mathrm{Sq}}^{0}(x),\operatorname{\mathrm{Sq}}^{0}(\operatorname{\mathrm{Sq}}^{0}(x)),\dots\}$ by iterating $\operatorname{\mathrm{Sq}}^{0}$. We say that a $\operatorname{\mathrm{Sq}}^{0}$-family is non-trivial if all members of the family are non-zero.

Example 1.18**.**

On the Adams 1-line the Hopf classes $\{h_{j}\}_{j\geq 0}$ form a $\operatorname{\mathrm{Sq}}^{0}$-family as

[TABLE]

Similarly, because $\operatorname{\mathrm{Sq}}^{0}$ is a ring map, we have $\operatorname{\mathrm{Sq}}^{0}$-families $\{h_{j}^{2}\}_{j\geq 0}$ and $\{h_{j}^{3}\}_{j\geq 0}$ as well.

Example 1.19**.**

The classes $g_{j}\in\operatorname{\mathrm{Ext}}^{4,2^{j+3}+2^{j+2}}_{\operatorname{\mathcal{A}}}(\mathbb{F}_{2},\mathbb{F}_{2})$ in the statement of Theorem 1.11 also form a $\operatorname{\mathrm{Sq}}^{0}$-family. In fact, the $g_{j}$ are defined to be the classes in the $\operatorname{\mathrm{Sq}}^{0}$-family generated by $g_{1}\coloneqq g$.

In 1995, Minami [Min95] made the following conjecture regarding $\operatorname{\mathrm{Sq}}^{0}$-families:

Conjecture 1.20** (New Doomsday Conjecture).**

For any $\operatorname{\mathrm{Sq}}^{0}$-family $\{x_{j}\}_{j\geq 0}$ on the Adams $E_{2}$-page only finitely many classes survive to the $E_{\infty}$-page.

Adams’s solution of the Hopf invariant one problem (Theorem 1.2) shows that the New Doomsday Conjecture is true on the Adams 1-line. Similarly, Hill–Hopkins–Ravenel’s solution of the Kervaire invariant one problem (Theorem 1.5) was the last, hardest, case of the New Doomsday Conjecture on the Adams 2-line.

Remark 1.21**.**

On the Adams 3-line and above, Conjecture 1.20 remains open. With our Theorem 1.11, the remaining open cases are the $\operatorname{\mathrm{Sq}}^{0}$-families $\{h_{j}^{2}h_{j+k+1}+h_{j+1}h_{j+k}^{2}\}_{j\geq 0}$ for $k\geq 2$.

Based on the uniformity of the differentials in Theorems 1.2 and 1.11 we propose the following refinement of Minami’s conjecture:

Conjecture 1.22** (Uniform Doomsday Conjecture).**

Let $\{a_{j}\}_{j\geq 0}$ be a non-trivial $\operatorname{\mathrm{Sq}}^{0}$-family on the Adams $E_{2}$-page. Then, there exists another $\operatorname{\mathrm{Sq}}^{0}$-family $\{b_{j}\}_{j\geq 0}$, an $r\geq 2$ and a class $c$ such that

[TABLE]

for $j\gg 0$.

Example 1.23**.**

Bruner’s formulas for power operations on Adams differentials naturally produces families of differentials of the form predicted by 1.22. For example one obtains families of differentials

1. (1)

$d_{2}(h_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(h_{j})=h_{0}h_{j}^{2}$, 2. (2)

$d_{2}(h_{j+1}^{2})=h_{0}\cdot\mathrm{Sq}^{1}(h_{j}^{2})=0$, 3. (3)

$d_{2}(c_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(c_{j})=h_{0}f_{j}$, 4. (4)

$d_{2}(d_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(d_{j})=0$, 5. (5)

$d_{2}(e_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(e_{j})=h_{0}x_{j}$, 6. (6)

$d_{2}(f_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(f_{j})=0$, 7. (7)

$d_{2}(g_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(g_{j})=0$, 8. (8)

$d_{2}(p_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(p_{j})=h_{0}h_{j}p^{\prime}_{j}$, 9. (9)

$d_{2}(D_{3}(j+1))=h_{0}\cdot\mathrm{Sq}^{1}(D_{3}(j))=h_{0}K_{j}$, 10. (10)

$d_{2}(p^{\prime}_{j+1})=h_{0}\cdot\mathrm{Sq}^{1}(p^{\prime}_{j})=h_{0}T_{j}$,

for $j\geq 1$.

Prior to the present work, all known cases of 1.22 followed as corollaries of Bruner’s power operation formulas and the authors’ core motivation in undertaking this project was to provide a more substantiative test of this conjecture. The differentials of 1.11 cannot be obtained from Bruner’s formulas. In fact, reversing the flow of information, the differentials of 1.11 can be interpreted as the first computation of an infinite family of hidden power operations in the Adams spectral sequence for the sphere.

1.1. An outline of the paper

In Section 2 we review the Miller square, motivic homotopy theory and set up notation for many of the spectral sequences we will use throughout the paper. In Section 3 we reduce the proof of 1.11 to a collection of propositions which we will verify across the remaining sections of the paper. In Section 4 we study the $E_{2}$-page of the Cartan–Eilenberg spectral sequence near $h_{j}^{3}$. In Section 5 we show that there are no Cartan–Eilenberg differentials near $h_{j}^{3}$ and in particular it is at this point that we show that the target of the Adams differential in 1.11 is non-trivial. In Section 6 we prove the key family of algebraic Novikov differentials at the heart of our proof. In Section 7 we use $\operatorname{\mathbb{F}}_{2}$-synthetic homotopy theory to analyze the classes $h_{j}^{2}$, proving 1.13, 1.15 and the final input necessary for the proof of 1.11.

1.2. Notations and conventions

1. (1)

We write $S^{n}$ for the $n$-sphere in spectra,

$S^{s,w}$ for the $\mathbb{C}$-motivic $(s,w)$-sphere $\Sigma^{s-w}(\mathbb{G}_{m})^{\otimes w}$ and

$\operatorname{\mathbb{S}}^{k,s}$ for the $\operatorname{\mathbb{F}}_{2}$-synthetic $(k,s)$-sphere $\Sigma^{-s}\nu(S^{k+s})$. 2. (2)

The indices we use in the various spectral sequences we consider are as follows: In the classical Adams $E_{2}$-page $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{a,t}(\mathbb{F}_{2},\mathbb{F}_{2})$, we use $a$ for the Adams filtration and $t$ for the internal degree. In the motivic Adams $E_{2}$-page $\operatorname{\mathrm{Ext}}^{a,t,w}_{\operatorname{\mathcal{A}}^{\textup{mot}}}(\mathbb{F}_{2}[\tau],\mathbb{F}_{2}[\tau])$, we use $a$ for the Adams filtration and $t$ for the internal degree and $w$ for weight. Moreover, from the motivic Cartan–Eilenberg spectral sequence, every element in $\operatorname{\mathrm{Ext}}^{a,t,w}_{\operatorname{\mathcal{A}}^{\textup{mot}}}(\mathbb{F}_{2}[\tau],\mathbb{F}_{2}[\tau])$ has a Cartan–Eilenberg filtration, denoted by $k$, and an Adams–Novikov filtration $s$, satisfying that $a=s+k$. 3. (3)

In order to avoid notational collisions we write $\tau$ for the usual $\mathbb{C}$-motivic map $\tau$ and $\lambda$ for the $\operatorname{\mathbb{F}}_{2}$-synthetic map usually denoted $\tau$. 4. (4)

In the final pair of sections it becomes important to distinguish between several different classes connected with Kervaire invariant one. We use $h_{j}^{2}$ for the classes on the Adams $E_{2}$-page. We use $\vartheta_{j}$ for certain class on the Adams–Novikov $E_{2}$-page which map to $h_{j}^{2}$ under the Thom reduction map (see 6.6). We use $\Theta_{j}$ for the Kervaire invariant one classes in the stable homotopy groups of spheres. We use $\theta_{j}$ for a choice of class in the synthetic homotopy groups of $\operatorname{\mathbb{S}}/\!\lambda^{k}$ (which exists when $h_{j}^{2}$ survives to the $E_{k+1}$-page of the Adams spectral sequence). 5. (5)

All objects are $p$-complete unless otherwise noted. 6. (6)

Outside of Section 2 we work entirely at the prime $2$.

Acknowledgments

The authors would like to thank Hood Chatham, Mike Hopkins, Hana Jia Kong, Ishan Levy, Haynes Miller, Andy Senger and Guozhen Wang for helpful conversations regarding the content of this paper. During the course of this work, the first author was supported by NSF grant DMS 2202992. The second author is partially supported by NSF grant DMS 2105462.

2. The Miller square and motivic homotopy theory

In this section we review necessary background for the proof of our main theorem—Theorem 1.11. We begin by recalling the Miller square, introduced in [Mil81], which captures the interplay between the Adams spectral sequence and the Adams–Novikov spectral sequence. We then discuss the recent cofiber of $tau$ method developed by Gheorghe, Isaksen, Wang and the second author [GWX21, IWX20b, IWX20a] which uses the motivic stable homotopy category over $\mathbb{C}$ and the motivic Adams spectral sequence to categorify the Miller square.

2.1. The Miller square

The Adams spectral sequence and the Adams–Novikov spectral sequence are two of the most effective methods of computing the homotopy groups of the $p$-completed sphere spectrum, $S^{0}$. They are spectral sequences of the form:

[TABLE]

where $\operatorname{\mathcal{A}}$ is the mod $p$ dual Steenrod algebra and $\mathrm{BP}$ is the Brown–Peterson spectrum at the prime $p$. For degrees, $s$ is the homological degree and is referred as the Adams filtration (resp. the Adams–Novikov filtration), and $t$ is the internal degree.

It is important to understand connections between them. A first connection is given by the Thom reduction map $\mathrm{BP}\rightarrow\operatorname{\mathbb{F}}_{p}$, which induces a map of spectral sequences

[TABLE]

that preserves the $(s,t)$-degrees. However, a general homotopy class in $\pi_{*}S^{0}$ doesn’t usually have the same Adams filtration as the Adams–Novikov filtration. So this map is not very useful for comparison of the Adams filtration and the Adams–Novikov filtration of a surviving homotopy class—it only tells us the latter is less or equal to the former.

A fundamental connection is the Miller square. We have an algebraic Novikov spectral sequence converging to the Adams–Novikov $E_{2}$-page, and a Cartan–Eilenberg spectral sequence converging to the Adams $E_{2}$-page. It turns out the $E_{2}$-pages of these two algebraic spectral sequences are isomorphic.

The algebraic Novikov spectral sequence comes the filtration of powers of the augmentation ideal $I=(p,v_{1},v_{2},\cdots)\subset\mathrm{BP}_{*}$. It has the form:

[TABLE]

[TABLE]

where $s$ and $k$ are homological degrees, and $t^{\prime}$ is internal degree. In particular, in the Adams–Novikov gradings, all differentials in the algebraic Novikov spectral sequence look like Adams–Novikov $d_{1}$-differentials.

Let $\mathcal{P}$ be the sub-Hopf algebra of squares inside $\operatorname{\mathcal{A}}$

[TABLE]

and let $\mathcal{Q}$ be the quotient Hopf algebra, $\mathcal{Q}\cong\operatorname{\mathcal{A}}\otimes_{\mathcal{P}}\operatorname{\mathbb{F}}_{2}\cong\Lambda_{\operatorname{\mathbb{F}}_{2}}[\xi_{1},\xi_{2},\cdots]$. The Cartan-Eilenberg spectral sequence comes from the extension of Hopf algebras $\mathcal{P}\to\operatorname{\mathcal{A}}\to\mathcal{Q}$. It has the form:

[TABLE]

[TABLE]

where $s$ and $k$ are homological degrees, and $t$ is internal degree. In particular, in the Adams gradings, all differentials in the Cartan–Eilenberg spectral sequence look like Adams $d_{1}$-differentials. The Hopf algebra $\mathcal{Q}$ is cocommutative and primitively generated by the exterior classes $\xi_{i+1}$, therefore we have

[TABLE]

where $q_{i}$ corresponds to $[\xi_{i+1}]$ and has $(k,t)$ bidegree $(1,2^{i+1}-1)$.

We identify the $E_{2}$-pages of the Cartan–Eilenberg spectral sequence and the algebraic Novikov spectral sequence by using the isomorphism of Hopf algebroids

[TABLE]

and the associated isomorphism of $\mathcal{P}$-comodule algebras

[TABLE]

where the middle isomorphisms identifies $q_{i}$ with $v_{i}$. Taking into account that $q_{i}$ has $(s,k,t)$-degree $(0,1,2^{i+1}-1)$ and $v_{i}$ has $(s,k,t^{\prime})$-degree $(0,1,2^{i+1}-2)$ the isomorphisms above provide an isomorphism

[TABLE]

by sending $s$ to $s$, $k$ to $k$, and $t$ to $t^{\prime}+k$. Altogether, we have introduced the Miller square:

[TABLE]

Remark 2.1**.**

Miller’s original motivation for studying his square was in order to deduce $d_{2}$-differentials in the Adams spectral sequence from $d_{2}$-differentials in the algebraic Novikov spectral sequence (see [Mil81]). Using motivic homotopy theory, one can generalize this method to obtain information about $d_{r}$-differentials for $r\geq 2$. We will explain this in the next subsection.

Remark 2.2**.**

At odd primes, the dual Steenrod algebra admits an additional Cartan grading which places $\xi_{i}$ in degree [math] and $\tau_{i}$ in degree $1$. As the differentials in the Cartan–Eilenberg spectral sequence must respect the Cartan grading, this spectral sequence collapses at the $E_{2}$-page.

2.2. Motivic homotopy theory

We work with the motivic stable homotopy category over $\mathbb{C}$. For the gradings, we denote by $S^{1,0}$ the simplicial sphere, and by $S^{1,1}$ the multiplicative group $\mathbb{G}_{m}=\mathbb{A}^{1}-0$. We use the same notation for their suspension spectra. We denote by $\operatorname{\mathbb{F}}_{p}^{\textup{mot}}$ the mod $p$ motivic Eilenberg-Mac Lane spectrum that represents the mod $p$ motivic cohomology, and by BPGL the motivic Brown-Peterson spectrum at the prime $p$. When the context is clear, we abuse notation and also write $S^{n,w}$ for the $\operatorname{\mathbb{F}}_{p}^{\textup{mot}}$-completed motivic sphere spectrum in bidegree $(n,w)$.

There is a map

[TABLE]

that induces a nonzero map on mod $p$ motivic homology. We denote by $S^{0,0}/\tau$ the cofiber of $\tau$.

[TABLE]

There is a Betti Realization functor $\mathbf{Re}$ from the motivic stable homotopy category over $\mathbb{C}$ to the classical stable homotopy category. We have

[TABLE]

The motivic dual Steenrod algebra over $\operatorname{\mathcal{C}}$ is a Hopf algebra over $\pi_{*,*}(\operatorname{\mathbb{F}}_{p}^{\textup{mot}})=\operatorname{\mathbb{F}}_{p}[\tau]$ which takes the following form (see [Voe03, Sec. 12])

[TABLE]

[TABLE]

We then have the motivic Adams spectral sequence ([DI10]) of the form

[TABLE]

[TABLE]

The quotient map $S^{0,0}\rightarrow S^{0,0}/\tau$ induces a map of the motivic Adams spectral sequences.

[TABLE]

The following theorem is crucial in the computation of classical and motivic stable homotopy groups of spheres over $\mathbb{C}$.

Theorem 2.3** ([GWX21, Theorem 1.17]).**

There is an isomorphism of tri-graded spectral sequences between the motivic Adams spectral sequence for $S^{0,0}/\tau$ and the algebraic Novikov spectral sequence.

[TABLE]

Remark 2.4**.**

Note that the $\operatorname{\mathrm{Ext}}$-groups in the left column in Theorem 2.3 are only defined when $t^{\prime}$ is odd, meanwhile in the right column the $\operatorname{\mathrm{Ext}}$-groups vanish for $t^{\prime}$ odd for sparsity reasons.

As explained in [GWX21, Subsection 1.3], the motivic deformation and the naturality of the Adams spectral sequences give us a zig-zag diagram.

[TABLE]

The right-hand horizontal maps are induced by the quotient map $S^{0,0}\rightarrow S^{0,0}/\tau$, and the left-hand horizontal maps are given by the Betti realization functor. The diagram of spectral sequences allow us to build up connections between the differentials in the classical Adams spectral sequence and the algebraic Novikov spectral sequence (by Theorem 2.3) through the motivic world.

Remark 2.5**.**

It is often useful to combine the isomorphisms in the Miller square and in Theorem 2.3 and obtain the following isomorphism between the $E_{2}$-pages of the Cartan-Eilenberg spectral sequence and the motivic Adams spectral sequence for $S^{0,0}/\tau$:

[TABLE]

We also have the motivic Adams-Novikov spectral sequence ([HKO11, Isa19]) of the form.

[TABLE]

[TABLE]

In [Isa19, Sections 6.1, 6.2], Isaksen proves the following rigidity theorem.

Theorem 2.6**.**

After a re-grading, the motivic Adams-Novikov spectral sequence for $\pi_{*,*}S^{0,0}$ is isomorphic to a $\tau$-Bockstein spectral sequence.

Finally, for some of our later arguments we need a motivic version of the Cartan–Eilenberg spectral sequence. Notably, this spectral sequence satisfies a rigidity theorem analogous to 2.6.

Construction 2.7**.**

We define

[TABLE]

The associated extension of Hopf algebroids $\mathcal{P}^{\textup{mot}}\rightarrow\operatorname{\mathcal{A}}^{\textup{mot}}\rightarrow\mathcal{Q}^{\textup{mot}}$ gives us a motivic version of the classical Cartan-Eilenberg spectral sequence. It has the form

[TABLE]

[TABLE]

Note that in this spectral sequence all $d_{r}$-differentials look like motivic Adams $d_{1}$-differential in the tri-gradings.

Theorem 2.8**.**

After a re-grading, the motivic Cartan-Eilenberg spectral sequence for $\operatorname{\mathrm{Ext}}_{{\operatorname{\mathcal{A}}}^{\textup{mot}}}^{*,*,*}(\operatorname{\mathbb{F}}_{p}[\tau],\operatorname{\mathbb{F}}_{p}[\tau])$ is isomorphic to a $\tau$-Bockstein spectral sequence.

Proof.

We have

[TABLE]

Here $\tau$ has $(s,k,t,w)$-degrees $(0,0,0,-1)$, and every element in $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{s,t}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}^{k}_{\mathcal{Q}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$ satisfies $w=\frac{t-k}{2}$ by Remark 2.5. Note that by sparseness this group is nonzero only if $t-k$ is even.

Consider $S^{0,0}/\tau$, we have the $E_{2}$-page of its MCESS isomorphic to $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{s,t}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}^{k}_{\mathcal{Q}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$. Since all differentials in MCESS preserve the $w$ and $t$-degrees, it must preserve the $k$-degree in the case for $S^{0,0}/\tau$, so its MCESS collapses at the $E_{2}$-page.

Back to $S^{0,0}$, again due to degree reasons, all $d_{r}$-differentials have the form

[TABLE]

and $d_{2n}=0$. One can check that there are no $\tau$-extensions in MCESS. This is precisely saying that after a regrading, the MCESS for $\operatorname{\mathrm{Ext}}^{*,*,*}_{\operatorname{\mathcal{A}}^{\textup{mot}}}(\operatorname{\mathbb{F}}_{2}[\tau],\operatorname{\mathbb{F}}_{2}[\tau])$ is isomorphic to a $\tau$-Bockstein spectral sequence and completes the proof. ∎

Remark 2.9**.**

The important consequence of Theorem 2.6 is that, the differentials in the classical Adams-Novikov spectral sequence completely determine the differentials in the motivic Adams-Novikov spectral sequence, and vice versa. Similarly, our Theorem 2.8 tells us that the differentials in the classical and motivic Cartan-Eilenberg spectral sequences determine each other.

Corollary 2.10**.**

Suppose there are no non-trivial Cartan–Eilenberg differentials entering tridegrees $(s,k,t)=(s,*,t)$. Let $\{\overline{x}_{i}\}_{i\in I}$ be a collection of generators of the permanent cycles on the Cartan–Eilenberg $E_{2}$-page in tridegree $(s,k,t)=(s,*,t)$. Then, we can lift each $\overline{x}_{i}$ to a class $x_{i}$ on the $E_{2}$-page of the motivic Adams spectral sequence for $S^{0,0}$ and any choice of lifts $\{x_{i}\}_{i\in I}$ provides a basis for $E_{2}^{s,*,t}$ as a free $\operatorname{\mathbb{F}}_{2}[\tau]$-module.

3. Proof of the main theorem

In this section, we give the proof of our main theorem, 1.11, modulo three inputs from the later sections of the paper. Our core strategy is to use the motivic zig-zag to gain information about the classical Adams $d_{4}$ differential on $h_{j}^{3}$ from the associated algebraic Novikov $d_{4}$ differential.

Recollection 3.1**.**

As in the classical Steenrod algebra, $\xi_{1}^{2^{j}}$ is a primitive element in the motivic dual Steenrod algebra. We let $h_{j}$ denote the associated class on the $E_{2}$-page of the motivic Adams spectral sequence. $h_{j}$ lives in $(a,s,t,w)$-degree $(1,\ 1,\ 2^{j},\ 2^{j-1})$. Similarly, we also denote the induced classes on the $E_{2}$-pages of the classical Adams sseq and motivic Adams sseq for $S^{0,0}/\tau$ by $h_{j}$.

In fact, the main result of this section is a computation of $d_{4}(h_{j}^{3})$ in the motivic Adams spectral sequence (with 1.11 being obtained from this by Betti realization). Before proceeding let us summarize what we need from later sections.

• •

In 3.2 we describe the $E_{2}$-pages of all three spectral sequences in the motivic zig-zag (classical Adams, motivic Adams, motivic Adams for $S^{0,0}/\tau$) in a neighborhood of the classes $h_{j}^{3}$. This theorem is proved in Sections 4 and 5 and summarized in Figure 1.

• •

In 3.4 we compute the differentials on $h_{j}^{3}$ in the motivic Adams sseq for $S^{0,0}/\tau$. This theorem is proved in Section 6.

• •

In 3.5 we give partial information on the classical Adams differentials on $h_{j}^{3}$. This theorem is proved in Section 7 using $\operatorname{\mathbb{F}}_{2}$-synthetic homotopy theory.

Theorem 3.2**.**

Let $j\geq 6$.

1. (1)

The $E_{2}$-page of the classical Adams spectral sequence for $S^{0}$,

[TABLE]

takes the following form near $h_{j}^{3}$:

[TABLE]

with the exception that in the case $j=6$ degree $(a,t-s)=(5,3\cdot 2^{6}-4)$ contains the additional class $h_{7}D_{3}(0)$, and that in the case $j=7$ degree $(a,t-s)=(5,3\cdot 2^{7}-3)$ contains the additional class $h_{8}D_{3}(1)$. 2. (2)

The $E_{2}$-page of the motivic Adams spectral sequence for $S^{0,0}$,

[TABLE]

takes the following form near $h_{j}^{3}$:

[TABLE]

where each of the generators $g_{j-2},h_{0}g_{j-2},h_{0}^{2}g_{j-2},h_{0}^{3}g_{j-2},h_{j}^{3},h_{0}h_{j}^{3},h_{0}^{2}h_{j}^{3}$ has weight $w=3\cdot 2^{j-1}$ and $h_{1}g_{j-2}$ has weight $w=3\cdot 2^{j-1}+1$, with the exception that in the case $j=6$ degree $(a,t-a)=(5,3\cdot 2^{6}-4)$ contains an additional $\operatorname{\mathbb{F}}_{2}[\tau]$ summand with generator $h_{7}D_{3}(0)$, and that in the case $j=7$ degree $(a,t-a)=(5,3\cdot 2^{7}-3)$ contains an additional $\operatorname{\mathbb{F}}_{2}[\tau]$ summand with generator $h_{8}D_{3}(1)$ of weight $w=3\cdot 2^{6}+1$. 3. (3)

The $E_{2}$-page of the motivic Adams spectral sequence for $S^{0,0}/\tau$,

[TABLE]

takes the following form near $h_{j}^{3}$:

[TABLE]

where the generators have the same weights as in (2), with the exception that in the case $j=6$ degree $(a,2w-t+a,t-a)=(5,4,3\cdot 2^{6}-4)$ possibly contains an additional class $h_{7}D_{3}(0)$, and that in the case $j=7$ degree $(a,2w-t+a,t-a)=(5,5,3\cdot 2^{7}-3)$ contains an additional class $h_{8}D_{3}(1)$. 4. (4)

Under the Betti realization and reduction mod $\tau$ maps each generator maps to the generator of the same name.

{sseqdata}

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\class

circle, inner sep=0.5pt, "?", font = \classcircle, inner sep=0.5pt, "?", font =

\class

fill, "g_j-2" below \classfill \structline\classfill \structline\classfill \structline

\class

fill \structline(-4,4)

\class

fill, "h_j^3" below \classfill \structline\classfill \structline

{sseqdata}

[ name = 3charts2hj3, xscale=1.5, yscale=1.5, x range = -4-3, y range = 27, x tick step = 1, y tick step = 1, grid = crossword, Adams grading, lax degree]

\class

rectangle, inner sep=1.5pt, "?", font = \classrectangle, inner sep=1.5pt, "?", font =

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\class

rectangle, fill, inner sep=3pt \structline(-4,4)

\class

rectangle, fill, inner sep=3pt, "h_j^3" below \classrectangle, fill, inner sep=3pt \structline\classrectangle, fill, inner sep=3pt \structline

{sseqdata}

[ name = 3charts3hj3, xscale=1.5, yscale=1.5, x range = -4-3, y range = 27, x tick step = 1, y tick step = 1, grid = crossword, Adams grading, lax degree]

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fill \structline(-4,4)

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circle, inner sep=0.5pt, "5", font =

\class

fill, "h_j^3" below \classfill \structline\classfill \structline\classfill, red \structline

As an amplification of the first four lines of the table in 3.2(2) we have the following corollary.

Corollary 3.3**.**

The Betti realization map from the $E_{2}$-page of the motivic Adams sseq for $S^{0,0}$ to the $E_{2}$-page of the Adams sseq for $S^{0}$ is injective in the tridegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$, $h_{0}^{3}g_{j-2}$ and their $\tau$-multiples.

Theorem 3.4**.**

Let $j\geq 6$. In the motivic Adams sseq for $S^{0,0}/\tau$,

1. (1)

$d_{2}(h_{j}^{3})=0$, 2. (2)

$d_{3}(h_{j}^{3})=0$* and* 3. (3)

$d_{4}(h_{j}^{3})=h_{0}^{3}g_{j-2}$.

Theorem 3.5**.**

Let $j\geq 6$. In the classical Adams sseq for $S^{0}$,

1. (1)

$d_{2}(h_{j}^{3})=0$, 2. (2)

$d_{3}(h_{j}^{3})$* is either [math] or $h_{0}^{2}g_{j-2}$ and* 3. (3)

$d_{4}(h_{j}^{3})$* is either [math] or $h_{0}^{3}g_{j-2}$ (if defined).*

With all our inputs ready we now begin proving our main theorem.

Lemma 3.6**.**

Let $j\geq 6$. In the motivic Adams sseq for $S^{0,0}/\tau$, there are no non-zero $d_{2}$ or $d_{3}$-differentials entering the tridegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$ or $h_{0}^{3}g_{j-2}$.

Proof.

Using 3.2(3) we can read off that the only potential sources for a $d_{2}$ or $d_{3}$-differential entering the tridegree of one of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$ or $h_{0}^{3}g_{j-2}$ are $h_{j}^{3}$, $h_{0}h_{j}^{3}$ and $h_{0}^{2}h_{j}^{3}$. The lemma now follows as a corollary of 3.4(1,2). ∎

Lemma 3.7**.**

Let $j\geq 6$.

1. (1)

In the motivic Adams sseq for $S^{0,0}$, $d_{2}(h_{j}^{3})=0$. 2. (2)

For $j\neq 7$, in the Adams sseq for $S^{0}$, there are no non-zero $d_{2}$-differentials entering the bidegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$ and $h_{0}^{3}g_{j-2}$. 3. (3)

For $j\neq 7$, in the motivic Adams sseq for $S^{0,0}$, there are no non-zero $d_{2}$-differentials entering the tridegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$, $h_{0}^{3}g_{j-2}$ or their $\tau$-multiples. 4. (4)

The Betti realization map from the $E_{3}$-page of the motivic Adams sseq for $S^{0,0}$ to the $E_{3}$-page of the Adams sseq for $S^{0}$ is injective in the tridegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$, $h_{0}^{3}g_{j-2}$ and their $\tau$-multiples.

Proof.

Using 3.3 we can deduce (1) from 3.5(1). For $j\neq 7$ we can read off from 3.2(1) that the only potential sources for Adams differential entering the bidegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$ and $h_{0}^{3}g_{j-2}$ are the classes $h_{j}^{3}$, $h_{0}h_{j}^{3}$, $h_{0}^{2}h_{j}^{3}$ and $h_{1}g_{j-2}$. Thus, (2) follows from 3.5(1) and 1.23 (which tells us that $d_{2}(g_{j-2})=0$). The injectivity from 3.3 implies that any entering motivic $d_{2}$-differential would induce an entering classical $d_{2}$-differential, therefore (2) implies (3). For $j\neq 7$, (4) is obtained by combining (2), (3) and 3.3.

For the $j=7$ case of (4) we observe that the additional generator $h_{8}D_{3}(1)$ has degree $(a,t,w)=(5,3\cdot 2^{7}+2,3\cdot 2^{6}+1)$ and that in order for this differential to create $\tau$-torsion on the motivic Adams $E_{3}$-page its target must be $\tau$-divisible. This would force us to have a non-trivial class in degree $(a,t,w)=(7,3\cdot 2^{7}+3,3\cdot 2^{6}+2)$. On the other hand, such a class would contradict the fact that the motivic Adams $E_{2}$-page vanishes when $t<2w$. ∎

Lemma 3.8**.**

Let $j\geq 6$.

1. (1)

In the classical Adams sseq for $S^{0}$, $d_{3}(h_{j}^{3})=0$. 2. (2)

In the motivic Adams sseq for $S^{0,0}$, $d_{3}(h_{j}^{3})=0$. 3. (3)

In the classical Adams sseq for $S^{0}$, there are no non-zero $d_{3}$-differentials entering the bidegrees of $h_{0}^{2}g_{j-2}$ and $h_{0}^{3}g_{j-2}$. 4. (4)

In the motivic Adams sseq for $S^{0,0}$, there are no non-zero $d_{3}$-differentials entering the tridegrees of $h_{0}^{2}g_{j-2}$, $h_{0}^{3}g_{j-2}$ or their $\tau$-multiples. 5. (5)

The Betti realization map from the $E_{4}$-page of the motivic Adams sseq for $S^{0,0}$ to the $E_{4}$-page of the Adams sseq for $S^{0}$ is injective in the tridegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$, $h_{0}^{3}g_{j-2}$ and their $\tau$-multiples.

Proof.

We begin with (2). The injectivity statement from 3.7(4) allows us to upgrade 3.5 to the claim that in the motivic Adams sseq for $S^{0,0}$, $d_{3}(h_{j}^{3})$ is either [math] or $h_{0}^{2}g_{j-2}$. Next we examine the reduction mod $\tau$ map to the motivic Adams sseq for $S^{0,0}/\tau$. From 3.2(3) and 3.6 we know that $h_{0}^{2}g_{j-2}$ is non-zero on the $E_{3}$-page of the motivic Adams sseq for $S^{0,0}/\tau$. Thus, (2) follows from 3.4(2) which tells us that $d_{3}(h_{j}^{3})=0$ in the motivic Adams sseq for $S^{0,0}/\tau$. (1) follows from (2) by Betti realization.

From 3.2(1) we can read off that the only potential sources for Adams differential entering the bidegrees of $h_{0}g_{j-2}$, $h_{0}^{2}g_{j-2}$ and $h_{0}^{3}g_{j-2}$ are the classes $h_{j}^{3}$ and $h_{0}h_{j}^{3}$. Thus, (3) follows from (1). The injectivity from 3.7(4) implies that any entering motivic $d_{3}$-differential would induce an entering classical $d_{3}$-differential, therefore (3) implies (4). (5) is obtained by combining (3), (4) and 3.7(4). ∎

Theorem 3.9**.**

Let $j\geq 6$. In the motivic Adams sseq for $S^{0,0}$ we have

[TABLE]

which Betti realizes to the non-trivial Adams differential of 1.11

Proof.

From Lemmas 3.7(1) and 3.8(2) we know that the class $h_{j}^{3}$ survives to the $E_{4}$-page of the motivic Adams sseq. Under the reduction mod $\tau$ map to $E_{4}$-page of the motivic Adams sseq for $S^{0,0}/\tau$ the class $d_{4}(h_{j}^{3})$ is sent to a non-zero class by 3.4(3) and 3.6, therefore $d_{4}(h_{j}^{3})\neq 0$. On the other hand, the injectivity statement from 3.8(4) allows us to upgrade 3.5(3) to the claim that in the motivic Adams sseq for the sphere $d_{4}(h_{j}^{3})$ is either [math] or $h_{0}^{3}g_{j-2}$. It follows that

[TABLE]

Using the injectivity from 3.8(4) again we obtain the desired non-trivial classical Adams $d_{4}$-differential from the motivic one. ∎

Remark 3.10**.**

One may prove directly that the element $h_{0}^{3}g_{j-2}$ survives to the classical Adams $E_{4}$-page, but it is not logically necessary for the proof of 3.9—3.9 in fact implies that $h_{0}^{3}g_{j-2}$ does not support a nonzero $d_{2}$ or $d_{3}$-differential.

4. The Cartan–Eilenberg $E_{2}$-page

Our goal in the next pair of sections is to prove 3.2. In this section we prove a weak form of 3.2(3). In fact, as explained in Section 2 the $E_{2}$-page of the motivic Adams sseq for $S^{0,0}/\tau$ is isomorphic to the $E_{2}$-page of the Cartan–Eilenberg sseq, therefore the main object of this section is the Cartan–Eilenberg $E_{2}$-page. In order to gain access to this $E_{2}$-page we construct an algebraic Atiyah–Hirzebruch sseq, converging to the Cartan-Eilenberg $E_{2}$-page. Then, in Section 5 we study the differentials in the Cartan–Eilenberg sseq and use this information to complete the proof of 3.2.

In order to motivate our strategy, let us focus on a particular claim from 3.2: the class $h_{0}^{3}g_{n+1}$ on the Adams $7$-line is non-trivial. On the $E_{2}$-page of the Cartan–Eilenberg sseq this class is detected by $q_{0}^{3}\cdot g_{n}$ and therefore it suffices for us to do two things (1) show that $q_{0}^{3}\cdot g_{n}$ is non-trivial on the Cartan–Eilenberg $E_{2}$-page and (2) show that $q_{0}^{3}\cdot g_{n}$ is not the target of a Cartan–Eilenberg differential. The class $q_{0}^{3}\cdot q_{n}$ is detected by a non-trivial class of the same name on the algebraic Atiyah–Hirzebruch $E_{1}$-page and we prove (1) by showing that this class is not the target of an algebraic Atiyah–Hirzebruch differential. We depict this strategy in the diagram of spectral sequences below.

[TABLE]

Notation 4.1**.**

In order to give names to elements on the $E_{2}$-page we use the injective map

[TABLE]

together with the Frobenius isomorphism $\mathcal{P}\cong\operatorname{\mathcal{A}}$ so that we can use the familiar names from $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}$. Note that this use of the Frobenius in naming classes means that the class $a$ on Cartan–Eilenberg $E_{2}$-page will detect the class $\operatorname{\mathrm{Sq}}^{0}(a)$ in $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}$. In particular, this means that the class $h_{j}^{3}$ on the motivic Adams $E_{2}$-page corresponds to the class $h_{j-1}^{3}$ on the Cartan–Eilenberg $E_{2}$-page.

Proposition 4.2**.**

Let $j\geq 5$. The $E_{2}$-page of the Cartan–Eilenberg spectral sequence takes the following form near $h_{j}^{3}$:

[TABLE]

As discussed in Section 2 we have an isomorphism $\operatorname{\mathrm{Ext}}_{\mathcal{Q}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})\cong\operatorname{\mathbb{F}}_{2}[q_{0},q_{1},\dots]$ where the polynomial generator $q_{i}$ has degree $(k,t)=(1,2^{i+1}-1)$. The $\mathcal{P}$-comodule structure on this polynomial algebra is given by

[TABLE]

[TABLE]

(see [Rav86, Theorem 4.3.3]). For further discussion of the Cartan–Eilenberg sseq see [AM17, Mil81, Rav86].

4.1. An algebraic Atiyah-Hirzebruch spectral sequence

In this subsection we construct an algebraic Atiyah–Hirzebruch spectral sequence which converges to the Cartan–Eilenberg $E_{2}$-page. The algAH sseq is the main tool we use in our proof of 4.2.

Construction 4.3**.**

The $\mathcal{P}$-comodule $\operatorname{\mathbb{F}}_{2}[q_{0},\dots]$ can be described as the free polynomial algebra on the $\mathcal{P}$-comodule $\operatorname{\mathbb{F}}_{2}\{q_{i}\}_{i\geq 0}$. We place an increasing filtration on $\operatorname{\mathbb{F}}_{2}\{q_{i}\}_{i\geq 0}$ where $q_{i}$ is in filtration $i$. Passing to polynomial algebras we obtain a filtered commutative algebra in $\mathcal{P}$-comodules whose underlying object is $\operatorname{\mathbb{F}}_{2}[q_{0},\dots]$. Associated to this filtration is a multiplicative spectral sequence computing $\operatorname{\mathrm{Ext}}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}[q_{0},\dots])$ which we will call the algebraic Atiyah–Hirzebruch spectral sequence.

Lemma 4.4**.**

The associated graded of the filtration on $\operatorname{\mathbb{F}}_{2}[q_{0},q_{1},\cdots]$ from 4.3 is given by the graded $\mathcal{P}$-comodule algebra $\operatorname{\mathbb{F}}_{2}[q_{0},q_{1},\cdots]$ with trivial $\mathcal{P}$-comodule structure where the monomial $q_{i_{1}}q_{i_{2}}\cdots q_{i_{k}}$ lives in filtration $i_{1}+i_{2}+\cdots+i_{k}$.

Proof.

The associated graded of a polynomial algebra on a filtered $P$-comodule $M$ is the polynomial algebra on the associated graded of $M$. Therefore, it suffices for us to observe that since we used the cellular filtration on $\operatorname{\mathbb{F}}_{2}\{q_{i}\}_{i\geq 0}$ the associated graded has trivial $P$-comodule structure. ∎

As a consequence of 4.4 the algAH sseq of 4.3 has signature

[TABLE]

where we give classes in $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{s,t}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$ degree $(s,k,t,i)=(s,0,t,0)$ and $q_{n}$ has degree $(s,k,t,i)=(0,1,2^{n+1}-1,n)$.

Remark 4.5**.**

The multiplicative structure on the algAH sseq coming from its construction via a filtered commutative algebra includes compatibility of the product structure on the $E_{1}$-page with products in $\operatorname{\mathrm{Ext}}_{P}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}[q_{0},q_{1},\dots])$ and a Liebniz rule for differentials.

Remark 4.6**.**

The $E_{1}$-page of the algAH sseq has a basis of elements of the form $q_{i_{1}}q_{i_{2}}\cdots q_{i_{k}}\cdot a$, where $a\in\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$. Furthermore, since the $k$-degree just records the number of $q$’s, algAH differentials preserve the number of $q$’s.

In general, the differentials in algAH sseq can be computed by embedding into the cobar complex that computes the $E_{2}$-page of Cartan–Eilenberg sseq. So in particular, the primary algAH differentials can be computed by using the $\mathcal{P}$-comodule structure map $\psi$.

Example 4.7**.**

We have

[TABLE]

Since $\xi_{1}$ detects $h_{0}$ in $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{1,1}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$, we have $\xi_{1}^{2}$ detects $h_{0}$ in $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{1,2}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$ (in the naming convention of 4.1). This gives us algAH differentials

[TABLE]

for $a\in\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$.

Example 4.8**.**

We have

[TABLE]

Examining the top AH filtration terms we obtain

[TABLE]

If $h_{1}a=0$ in $\operatorname{\mathrm{Ext}}_{P}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$, then we have $d_{1}(q_{2}\cdot a)=0$. In this case, we have

[TABLE]

This is due to the fact that in cobar complex for $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$, the nullhomotopy of $h_{0}h_{1}$ is $\xi_{2}$. Note that the indeterminacy of the set $q_{0}\cdot\langle h_{0},h_{1},a\rangle$ is killed by $d_{1}$-differentials.

Example 4.9**.**

[TABLE]

therefore we have

[TABLE]

Generalizing, these examples we have the following lemma.

Lemma 4.10**.**

We have the following differentials in algAHSS for $n\geq 0$.

1. (1)

$d_{1}(q_{n+1}\cdot a)=q_{n}\cdot h_{n}a$. 2. (2)

$d_{2}(q_{n+1}^{2}\cdot a)=q_{n}^{2}\cdot h_{n+1}a$. 3. (3)

$d_{2}(q_{n+2}\cdot a)=q_{n}\cdot\langle h_{n},h_{n+1},a\rangle$, if $h_{n+1}a=0$. 4. (4)

$d_{4}(q_{n+2}^{2}\cdot a)=q_{n}^{2}\cdot\langle h_{n+1},h_{n+2},a\rangle$, if $h_{n+2}a=0$. 5. (5)

$d_{3}(q_{n+3}\cdot a)=q_{n}\cdot\langle h_{n},h_{n+1},h_{n+2},a\rangle$, if $h_{n+2}a=0$ and $0\in\langle h_{n+1},h_{n+2},a\rangle$. 6. (6)

$d_{6}(q_{n+3}^{2}\cdot a)=q_{n}^{2}\cdot\langle h_{n+1},h_{n+2},h_{n+3},a\rangle$, if $h_{n+3}a=0$ and $0\in\langle h_{n+2},h_{n+3},a\rangle$.

Proof.

The differentials in $(1)$ and $(2)$ follow from the comodule structure map $\psi$ modulo elements in lower AH filtration

[TABLE]

[TABLE]

and the facts that $\xi_{1}^{2^{n+1}}$ and $\xi_{1}^{2^{n+2}}$ detects $h_{n}$ and $h_{n+1}$ in the cobar complex that computes $\operatorname{\mathrm{Ext}}_{P}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$.

The differentials in $(3)$ and $(4)$ follow from the comodule structure map $\psi$ modulo elements in lower AH filtration

[TABLE]

[TABLE]

and the facts that $\xi_{2}^{2^{n+1}}$ and $\xi_{2}^{2^{n+2}}$ detects the nullhomotopy of $h_{n}h_{n+1}$ and $h_{n+1}h_{n+2}$ in the cobar complex that computes $\operatorname{\mathrm{Ext}}_{P}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$.

Similarly, the elements $\xi_{3}^{2^{n+1}}$ and $\xi_{3}^{2^{n+2}}$ detects the null homotopy of the Massey products $\langle h_{n},h_{n+1},h_{n+2}\rangle$ and $\langle h_{n+1},h_{n+2},h_{n+3}\rangle$, and the differentials in $(5)$ and $(6)$ follow.

Note that indeterminacies of the expressions $q_{n}\cdot\langle h_{n},h_{n+1},a\rangle$, $q_{n}^{2}\cdot\langle h_{n+1},h_{n+2},a\rangle$,

$q_{n}\cdot\langle h_{n},h_{n+1},h_{n+2},a\rangle$, and $q_{n}^{2}\cdot\langle h_{n+1},h_{n+2},h_{n+3},a\rangle$ are killed by shorter differentials. ∎

4.2. The algebraic Atiyah–Hirzebruch $E_{1}$-page

In this subsection we compute the $E_{1}$-page of the algAH sseq in the degrees we will need for proving 4.2.

Recollection 4.11**.**

Recall that we have the following generators in the cohomology of the Steenrod algebra

[TABLE]

Lemma 4.12**.**

In degree $(s,k,t)=(3,3,3\cdot 2^{n+3}+3)$ the $E_{1}$-page of the algAH sseq consists of the following classes for $n\geq 4$,

1. (1)

$q_{0}q_{1}q_{n}\cdot c_{n}$, 2. (2)

$q_{0}^{3}\cdot h_{n+1}^{2}h_{n+3}=q_{0}^{3}\cdot h_{n+2}^{3}$, 3. (3)

$q_{0}^{2}q_{n+1}\cdot h_{0}h_{n+1}h_{n+3}$, 4. (4)

$q_{0}^{2}q_{n+3}\cdot h_{0}h_{n+1}^{2}$, 5. (5)

$q_{0}q_{n+1}^{2}\cdot h_{0}^{2}h_{n+3}$, 6. (6)

$q_{0}q_{n+1}q_{n+3}\cdot h_{0}^{2}h_{n+1}$, 7. (7)

$q_{n+1}^{2}q_{n+3}\cdot h_{0}^{3}$, 8. (8)

$q_{0}^{2}q_{n+2}\cdot h_{0}h_{n+2}^{2}$, 9. (9)

$q_{0}q_{n+2}^{2}\cdot h_{0}^{2}h_{n+2}$, 10. (10)

$q_{n+2}^{3}\cdot h_{0}^{3}$, 11. (11)

$q_{0}q_{1}q_{n+1}\cdot h_{n}^{2}h_{n+3}$, 12. (12)

$q_{0}q_{n}^{2}\cdot h_{1}h_{n+1}h_{n+3}$, 13. (13)

$q_{0}q_{n}q_{n+1}\cdot h_{1}h_{n}h_{n+3}$, 14. (14)

$q_{0}q_{n+1}q_{n+3}\cdot h_{1}h_{n}^{2}$, 15. (15)

$q_{1}q_{n}^{2}\cdot h_{0}h_{n+1}h_{n+3}$, 16. (16)

$q_{1}q_{n}q_{n+1}\cdot h_{0}h_{n}h_{n+3}$, 17. (17)

$q_{1}q_{n+1}q_{n+3}\cdot h_{0}h_{n}^{2}$, 18. (18)

$q_{0}q_{n+1}^{2}\cdot h_{1}h_{n+2}^{2}$, 19. (19)

$q_{0}q_{n+2}^{2}\cdot h_{1}h_{n+1}^{2}$, 20. (20)

$q_{1}q_{n+1}^{2}\cdot h_{0}h_{n+2}^{2}$, 21. (21)

$q_{1}q_{n+2}^{2}\cdot h_{0}h_{n+1}^{2}$.

Proof.

As discussed in subsection 4.1, the $E_{1}$-page of the algAH sseq in degree $(s,k,t)=(3,3,3\cdot 2^{n+3}+3)$ has a basis of elements of the form $q_{i}q_{j}q_{k}\cdot a$, where

[TABLE]

The classical Adams $3$-line $\operatorname{\mathrm{Ext}}^{3,*}_{\operatorname{\mathcal{A}}}$ is generated by the elements $c_{\ell}$ and $h_{a}h_{b}h_{c}$. So the candidates are of the following two forms

• •

$q_{i}q_{j}q_{k}\cdot c_{\ell}$,

• •

$q_{i}q_{j}q_{k}\cdot h_{a}h_{b}h_{c}$.

We may also assume that $i\leq j\leq k$ and $a\leq b\leq c$.

We start by considering classes of the form $q_{i}q_{j}q_{k}\cdot c_{\ell}$. The class $q_{i}q_{j}q_{k}\cdot c_{\ell}$ has $t$-degree

[TABLE]

from this we obtain the equation

[TABLE]

which can be simplified to the equation

[TABLE]

Since we are considering $n\geq 4$, the right hand side of $(*)$ is 3 mod 64, and is at least 195.

When $\ell=0$, the equation $(*)$ becomes

[TABLE]

Not possible for $n\geq 4$.

When $\ell=1$, we must have $i=0$, then equation $(*)$ becomes

[TABLE]

Not possible for $n\geq 4$.

When $\ell=2$, we must have $i=0,j=1$, then equation $(*)$ becomes

[TABLE]

Not possible for $n\geq 4$.

When $\ell\geq 3$, we must have $i=0,j=1$, then equation $(*)$ becomes

[TABLE]

We must have $k=\ell=n$, which is at least 4. This is case $(1):q_{0}q_{1}q_{n}\cdot c_{n}$.

Next we consider classes of the form $q_{i}q_{j}q_{k}\cdot h_{a}h_{b}h_{c}$. The class $q_{i}q_{j}q_{k}\cdot h_{a}h_{b}h_{c}$ has $t$-degree

[TABLE]

from this we obtain the equation

[TABLE]

which can be simplified to the equation

[TABLE]

Either two or three terms on the right hand side of $(**)$ contribute to $1+2$. So we only have the following 4 possibilities for the unordered set $\{2^{i},\ 2^{j},\ 2^{k},\ 2^{a},\ 2^{b},\ 2^{c}\}$.

• •

$\{1,\ 1,\ 1,\ 2^{n+1},\ 2^{n+1},\ 2^{n+3}\}$,

• •

$\{1,\ 1,\ 1,\ 2^{n+2},\ 2^{n+2},\ 2^{n+2}\}$,

• •

$\{1,\ 2,\ 2^{n},\ 2^{n},\ 2^{n+1},\ 2^{n+3}\}$,

• •

$\{1,\ 2,\ 2^{n+1},\ 2^{n+1},\ 2^{n+2},\ 2^{n+2}\}$.

For the set $\{1,\ 1,\ 1,\ 2^{n+1},\ 2^{n+1},\ 2^{n+3}\}$, we have candidates

[TABLE]

[TABLE]

These are the cases $(2)-(7)$.

For the set $\{1,\ 1,\ 1,\ 2^{n+2},\ 2^{n+2},\ 2^{n+2}\}$, we have candidates

[TABLE]

Note that we have a relation $h_{n+2}^{3}=h_{n+1}^{2}h_{n+3}$ in $\operatorname{\mathrm{Ext}}_{A}^{3,*}$ and hence in $\operatorname{\mathrm{Ext}}_{P}^{3,2*}$. These are the cases $(2)$ and $(8)-(10)$.

For the set $\{1,\ 2,\ 2^{n},\ 2^{n},\ 2^{n+1},\ 2^{n+3}\}$, we have candidates

[TABLE]

[TABLE]

Note that we have a relation $h_{n}h_{n+1}=0$ in $\operatorname{\mathrm{Ext}}_{A}^{2,*}$ and hence in $\operatorname{\mathrm{Ext}}_{P}^{2,2*}$. So certain candidates are already zero (e.g. $q_{0}q_{1}q_{n}\cdot h_{n}h_{n+1}h_{n+3}$). These are the cases $(11)-(17)$.

For the set $\{1,\ 2,\ 2^{n+1},\ 2^{n+1},\ 2^{n+2},\ 2^{n+2}\}$, we have candidates

[TABLE]

Again, due the relation $h_{n}h_{n+1}=0$, certain candidates are already zero so we don’t list them. These are the cases $(18)-(21)$.

This completes the discussion of elements of the form $q_{i}q_{j}q_{k}\cdot h_{a}h_{b}h_{c}$ and therefore the proof of this lemma. ∎

Lemma 4.13**.**

In degree $(s,k,t)=(1,5,3\cdot 2^{n+3}+3)$ the $E_{1}$-page of the algAHSS consists of the following classes for $n\geq 4$,

1. (1)

$q_{0}^{3}q_{n+2}q_{n+3}\cdot h_{0}$, 2. (2)

$q_{0}^{4}q_{n+3}\cdot h_{n+2}$, 3. (3)

$q_{0}^{4}q_{n+2}\cdot h_{n+3}$, 4. (4)

$q_{0}q_{1}q_{n+1}^{2}q_{n+3}\cdot h_{0}$, 5. (5)

$q_{0}^{2}q_{n+1}^{2}q_{n+3}\cdot h_{1}$, 6. (6)

$q_{0}^{2}q_{1}q_{n+1}q_{n+3}\cdot h_{n+1}$, 7. (7)

$q_{0}^{2}q_{1}q_{n+1}^{2}\cdot h_{n+3}$, 8. (8)

$q_{0}q_{1}q_{n+2}^{3}\cdot h_{0}$, 9. (9)

$q_{0}^{2}q_{n+2}^{3}\cdot h_{1}$, 10. (10)

$q_{0}^{2}q_{1}q_{n+2}^{2}\cdot h_{n+2}$, 11. (11)

$q_{1}q_{n}^{2}q_{n+1}q_{n+3}\cdot h_{1}$, 12. (12)

$q_{1}^{2}q_{n}q_{n+1}q_{n+3}\cdot h_{n}$, 13. (13)

$q_{1}^{2}q_{n}^{2}q_{n+3}\cdot h_{n+1}$, 14. (14)

$q_{1}^{2}q_{n}^{2}q_{n+1}\cdot h_{n+3}$, 15. (15)

$q_{1}q_{n+1}^{2}q_{n+2}^{2}\cdot h_{1}$, 16. (16)

$q_{1}^{2}q_{n+1}q_{n+2}^{2}\cdot h_{n+1}$, 17. (17)

$q_{1}^{2}q_{n+1}^{2}q_{n+2}\cdot h_{n+2}$, 18. (18)

$q_{n-1}^{2}q_{n}q_{n+1}q_{n+3}\cdot h_{2}$, 19. (19)

$q_{2}q_{n-1}q_{n}q_{n+1}q_{n+3}\cdot h_{n-1}$, 20. (20)

$q_{2}q_{n-1}^{2}q_{n+1}q_{n+3}\cdot h_{n}$, 21. (21)

$q_{2}q_{n-1}^{2}q_{n}q_{n+3}\cdot h_{n+1}$, 22. (22)

$q_{2}q_{n-1}^{2}q_{n}q_{n+1}\cdot h_{n+3}$, 23. (23)

$q_{n}^{4}q_{n+3}\cdot h_{2}$, 24. (24)

$q_{2}q_{n}^{3}q_{n+3}\cdot h_{n}$, 25. (25)

$q_{2}q_{n}^{4}\cdot h_{n+3}$, 26. (26)

$q_{n}^{2}q_{n+1}q_{n+2}^{2}\cdot h_{2}$, 27. (27)

$q_{2}q_{n}q_{n+1}q_{n+2}^{2}\cdot h_{n}$, 28. (28)

$q_{2}q_{n}^{2}q_{n+2}^{2}\cdot h_{n+1}$, 29. (29)

$q_{2}q_{n}^{2}q_{n+1}q_{n+2}\cdot h_{n+2}$, 30. (30)

$q_{n+1}^{4}q_{n+2}\cdot h_{2}$, 31. (31)

$q_{2}q_{n+1}^{3}q_{n+2}\cdot h_{n+1}$, 32. (32)

$q_{2}q_{n+1}^{4}\cdot h_{n+2}$.

Proof.

The $E_{1}$-page of the algAH sseq in degree $(s,k,t)=(1,5,3\cdot 2^{n+3}+3)$ has a basis of elements of the form

[TABLE]

where $i_{1}\leq i_{2}\leq i_{3}\leq i_{4}\leq i_{5}$.

Considering the $t$-degrees, we have an equation

[TABLE]

which can be simplified to the equation

[TABLE]

Up to four terms on the right hand side contribute to $4$. So we have the following 9 possibilities for the unordered set $\{2^{m},\ 2^{i_{1}},\ 2^{i_{2}},\ 2^{i_{3}},\ 2^{i_{4}},\ 2^{i_{5}}\}$.

• •

$\{1,\ 1,\ 1,\ 1,\ 2^{n+2},\ 2^{n+3}\}$,

• •

$\{1,\ 1,\ 2,\ 2^{n+1},\ 2^{n+1},\ 2^{n+3}\}$,

• •

$\{1,\ 1,\ 2,\ 2^{n+2},\ 2^{n+2},\ 2^{n+2}\}$,

• •

$\{2,\ 2,\ 2^{n},\ 2^{n},\ 2^{n+1},\ 2^{n+3}\}$,

• •

$\{2,\ 2,\ 2^{n+1},\ 2^{n+1},\ 2^{n+2},\ 2^{n+2}\}$,

• •

$\{4,\ 2^{n-1},\ 2^{n-1},\ 2^{n},\ 2^{n+1},\ 2^{n+3}\}$,

• •

$\{4,\ 2^{n},\ 2^{n},\ 2^{n},\ 2^{n},\ 2^{n+3}\}$,

• •

$\{4,\ 2^{n},\ 2^{n},\ 2^{n+1},\ 2^{n+2},\ 2^{n+2}\}$,

• •

$\{4,\ 2^{n+1},\ 2^{n+1},\ 2^{n+1},\ 2^{n+1},\ 2^{n+2}\}$.

The rest of proof works similarly to the proof of Lemma 4.12. ∎

Remark 4.14**.**

Note that since $q_{0}$ acts injectively on the algAH $E_{1}$-page we can read off the collection of elements on the $E_{1}$-page in degree $(s,k,t)=(3,2,3\cdot 2^{n+3}+2)$ by extracting the subset of elements divisible by $q_{0}$ from 4.12.

4.3. algebraic Atiyah–Hirzebruch differentials

In this subsection we complete the proof of 4.2 by computing the relevant degrees of the algAH sseq. Our computation of the algAH differentials uses multiplicative and Massey product structures in $\operatorname{\mathrm{Ext}}_{\mathcal{P}}$ together with descriptions of differentials in the algAH sseq from 4.10.

We begin with the following lemma which we will need in order to conclude that the targets of various algAH differentials are non-trivial.

Lemma 4.15**.**

The following elements are nonzero in $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{4,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$.

• •

$h_{0}c_{n}$,  for  $n\geq 2$,

• •

$h_{0}^{3}h_{n}$,  for  $n\geq 3$,

• •

$h_{0}h_{n}^{2}h_{n+3}$,  for  $n\geq 3$,

• •

$h_{1}h_{n}^{2}h_{n+3}$,  for  $n\geq 4$,

• •

$h_{1}h_{n}^{3}$,  for  $n\geq 5$,

• •

$h_{0}^{2}h_{n+1}h_{n+3}$,  for  $n\geq 1$,

• •

$h_{0}^{2}h_{n}h_{n+3}$,  for  $n\geq 2$,

• •

$h_{0}^{2}h_{n}^{2}$,  for  $n\geq 4$,

• •

$h_{1}c_{n}$,  for  $n\geq 3$.

Proof.

The complete description of $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{\leq 4,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$ is given as Theorems 1.2 and 1.3 in [Lin08]. In particular, all relations that are satisfied among the elements $h_{n}$ and $c_{n}$ up to the Adams filtration 4 are the following:

[TABLE]

[TABLE]

One checks that this lemma is true. ∎

Lemma 4.16**.**

• •

$\operatorname{\mathrm{Ext}}^{3,\ 3\cdot 2^{n+3}+3}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{3}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$* is either $\operatorname{\mathbb{F}}_{2}\{q_{0}^{3}\cdot h_{n+2}^{3}\}$ or [math].*

• •

$\operatorname{\mathrm{Ext}}^{4,\ 3\cdot 2^{n+3}+3}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{3}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$* contains $q_{0}^{3}\cdot g_{n}\neq 0$.*

• •

$\operatorname{\mathrm{Ext}}^{4,\ 3\cdot 2^{n+3}+2}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{2}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$* contains $q_{0}^{2}\cdot g_{n}\neq 0$.*

• •

$\operatorname{\mathrm{Ext}}^{4,\ 3\cdot 2^{n+3}+1}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{1}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$* contains $q_{0}\cdot g_{n}\neq 0$.*

Proof.

We begin by noting that $q_{0}^{3}\cdot g_{n}$, $q_{0}^{2}\cdot g_{n}$ and $q_{0}^{1}\cdot g_{n}$ are permanent cycles in the algAH sseq (since they are each a product of permanent cycles). In order to prove the second bullet point, we will show that $q_{0}^{3}\cdot g_{n}$ is not hit by an algAH differential. Note that if $q_{0}^{3}\cdot g_{n}$ is non-zero on the $E_{\infty}$-page of the algAH sseq, then so are $q_{0}^{2}\cdot g_{n}$ and $q_{0}\cdot g_{n}$. Therefore, the third and fourth bullet points will follow. We prove the first bullet point by showing that in the algAH sseq this degree has only $q_{0}^{3}\cdot h_{n+2}^{3}$ by the $E_{7}$-page.

In 4.12 we determined the $E_{1}$-page of the algAH sseq in degree $(s,\ k,\ t)=(3,\ 2,\ 3\cdot 2^{j}+2)$. What we must do now is show that for all 21 elements in 4.12, other than case $(2)$, each element either supports or is killed by a short algAH differential, so none of them can kill $q_{0}^{3}\cdot g_{n}$.

It is clear that being an element in $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})\{q_{0}^{3}\}$, $q_{0}^{3}\cdot h_{n+2}^{3}$ is a permanent cycle so it cannot kill $q_{0}^{3}\cdot g_{n}$. This accounts for the case $(2)$.

Using 4.10(1) and the Liebniz rule, we obtain the algAH $d_{1}$-differentials displayed in Figure 2.

For the cases $(1),(7),(10),(11),(13)-(17),(20),(21)$, by Lemma 4.15, the targets are all nonzero for $n\geq 4$. Note that in the case $(14)$ for $n=4$, the first term of the target $q_{0}q_{4}q_{7}\cdot h_{1}h_{4}^{3}$ is zero, while the second term $q_{0}q_{5}q_{6}\cdot h_{1}h_{4}^{2}h_{6}$ is nonzero. For $n\geq 5$, both terms are nonzero.

Using 4.10(2) and the Liebniz rule, we have the following algAH $d_{2}$-differentials.

[TABLE]

One observes that the sources and targets of the above two $d_{2}$-differentials survives to the $E_{2}$-page of the algAH sseq. In fact, for the case $(18)$, we have a $d_{1}$-differential.

[TABLE]

So in the $E_{2}$-page, we have a relation that $q_{0}q_{n+2}^{2}\cdot h_{1}h_{n+2}=q_{0}q_{n+1}q_{n+3}\cdot h_{1}h_{n+1}$. One may also use 4.10(3) to prove a $d_{2}$-differential

[TABLE]

which is equivalent to the above one.

Using 4.10(6), we have the following algAH $d_{6}$-differential

[TABLE]

One observes that all targets of the above algAH differentials are linearly independent. Since most of the targets only have one term, this is not hard to check. This completes the proof. ∎

Lemma 4.17**.**

$\operatorname{\mathrm{Ext}}^{3,\ 3\cdot 2^{n+3}+2}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{2}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$* is either $\operatorname{\mathbb{F}}_{2}\{q_{0}^{2}\cdot h_{n+2}^{3}\}$ or [math].*

Proof.

We consider all elements in the $E_{1}$-page of algAH sseq in degree $(s,\ k,\ t)=(3,\ 2,\ 3\cdot 2^{j}+2)$. In 4.14 we described how each such element can be obtained by dividing one of the elements from 4.12 by $q_{0}$. From this we obtain the following list of classes in degree $(s,\ k,\ t)=(3,\ 2,\ 3\cdot 2^{j}+2)$.

1. (1)

$q_{1}q_{n}\cdot c_{n}$, 2. (2)

$q_{0}^{2}\cdot h_{n+1}^{2}h_{n+3}=q_{0}^{2}\cdot h_{n+2}^{3}$, 3. (3)

$q_{0}q_{n+1}\cdot h_{0}h_{n+1}h_{n+3}$, 4. (4)

$q_{0}q_{n+3}\cdot h_{0}h_{n+1}^{2}$, 5. (5)

$q_{n+1}^{2}\cdot h_{0}^{2}h_{n+3}$, 6. (6)

$q_{n+1}q_{n+3}\cdot h_{0}^{2}h_{n+1}$, 7. (8)

$q_{0}q_{n+2}\cdot h_{0}h_{n+2}^{2}$, 8. (9)

$q_{n+2}^{2}\cdot h_{0}^{2}h_{n+2}$, 9. (11)

$q_{1}q_{n+1}\cdot h_{n}^{2}h_{n+3}$, 10. (12)

$q_{n}^{2}\cdot h_{1}h_{n+1}h_{n+3}$, 11. (13)

$q_{n}q_{n+1}\cdot h_{1}h_{n}h_{n+3}$, 12. (14)

$q_{n+1}q_{n+3}\cdot h_{1}h_{n}^{2}$, 13. (18)

$q_{n+1}^{2}\cdot h_{1}h_{n+2}^{2}$, 14. (19)

$q_{n+2}^{2}\cdot h_{1}h_{n+1}^{2}$.

Other than the cases $(2)$, $(5)$, $(6)$ and $(9)$, the differentials in the proof of 4.16 are $q_{0}$-divisible, therefore these candidates do not survive the algAH sseq. For the cases $(6)$ and $(9)$, we have an algAH $d_{1}$-differential

[TABLE]

and by 4.10(2), we have the following algAH $d_{2}$-differential.

[TABLE]

For case $(5)$, we have the following algAH $d_{2}$-differential by 4.10(2),

[TABLE]

This completes the proof. ∎

Lemma 4.18**.**

$\operatorname{\mathrm{Ext}}^{3,\ 3\cdot 2^{n+3}+1}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{1}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))$* is either $\operatorname{\mathbb{F}}_{2}\{q_{0}\cdot h_{n+2}^{3}\}$ or [math].*

Proof.

We consider all elements in the $E_{1}$-page of algAH sseq in degree $(s,\ k,\ t)=(3,\ 1,\ 3\cdot 2^{j}+2)$. As in 4.14 we can determine all such elements by beginning with the list from 4.12 and dividing by $q_{0}^{2}$.

1. (2)

$q_{0}\cdot h_{n+1}^{2}h_{n+3}=q_{0}\cdot h_{n+2}^{3}$, 2. (3)

$q_{n+1}\cdot h_{0}h_{n+1}h_{n+3}$, 3. (4)

$q_{n+3}\cdot h_{0}h_{n+1}^{2}$, 4. (8)

$q_{n+2}\cdot h_{0}h_{n+2}^{2}$,

For the cases $(3)$ and $(8)$, we have the following Atiyah-Hirzebruch $d_{1}$-differentials:

[TABLE]

For the case $(4)$, by 4.10(5), we have the following algAH $d_{3}$-differential

[TABLE]

This completes the proof. ∎

Lemma 4.19**.**

$\operatorname{\mathrm{Ext}}^{1,\ 3\cdot 2^{n+3}+3}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{5}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))\cong 0$.

Proof.

From 4.13 we know the $E_{1}$-page of the algebraic Atiyah–Hirzebruch spectral sequence in degree $(s,k,t)=(1,5,3\cdot 2^{n+3}+3)$. Using 4.10(1) we compute the relevant algAH $d_{1}$-differentials. These differentials are displayed in Figures 5 and 6. In order to pass the $E_{2}$-page we must also verify that the targets of these $d_{1}$-differentials are all linearly independent. For this we note that two classes $q_{i_{1}}q_{i_{2}}q_{i_{3}}q_{i_{4}}q_{i_{5}}\cdot a$ and $q_{j_{1}}q_{j_{2}}q_{j_{3}}q_{j_{4}}q_{j_{5}}\cdot b$ (with the $i$’s and $j$’s in non-decreasing order) on the $E_{1}$-page can only satisfy a relation if $i_{k}=j_{k}$ for $k=1,\dots,5$. Examining Figures 5 and 6 we see that no pair of classes satisfy this condition.

On the $E_{2}$-page of the algAH sseq only one of the 32 candidates from Lemma 4.13 is still present: (13). Using 4.10(2,3), we obtain the following $d_{2}$-differential.

[TABLE]

As the target is non-zero on the $E_{2}$-page of the algAH sseq this $d_{2}$-differential is non-zero. In particular, the algAH sseq is empty in degree $(s,k,t)=(1,5,3\cdot 2^{n+3}+3)$ starting from the $E_{3}$-page. ∎

Lemma 4.20**.**

$\operatorname{\mathrm{Ext}}^{1,\ 3\cdot 2^{n+3}+2}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{4}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))\cong 0$.

Proof.

Dividing by $q_{0}$ as in 4.14 we can determine the $E_{1}$-page of the algAH sseq in degree $(s,k,t)=(1,4,3\cdot 2^{n+3}+2)$ from 4.13. It contains the following classes:

1. (1)

$q_{0}^{2}q_{j}q_{j+1}\cdot h_{0}$, 2. (2)

$q_{0}^{3}q_{j+1}\cdot h_{j}$, 3. (3)

$q_{0}^{3}q_{j}\cdot h_{j+1}$, 4. (4)

$q_{1}q_{j-1}^{2}q_{j+1}\cdot h_{0}$, 5. (5)

$q_{0}q_{j-1}^{2}q_{j+1}\cdot h_{1}$, 6. (6)

$q_{0}q_{1}q_{j-1}q_{j+1}\cdot h_{j-1}$, 7. (7)

$q_{0}q_{1}q_{j-1}^{2}\cdot h_{j+1}$, 8. (8)

$q_{1}q_{j}^{3}\cdot h_{0}$, 9. (9)

$q_{0}q_{j}^{3}\cdot h_{1}$, 10. (10)

$q_{0}q_{1}q_{j}^{2}\cdot h_{j}$.

Using 4.10(1) we compute the relevant algAH $d_{1}$-differentials. These differentials are displayed in Figure 7. In particular, the algAH sseq is empty in degree $(s,k,t)=(1,4,3\cdot 2^{n+3}+2)$ starting from the $E_{2}$-page. ∎

Lemma 4.21**.**

$\operatorname{\mathrm{Ext}}^{1,\ 3\cdot 2^{n+3}+1}_{P}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{3}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))\cong 0$.

Proof.

Dividing by $q_{0}$ as in 4.14 again we can determine the $E_{1}$-page of the algAH sseq in degree $(s,k,t)=(1,3,3\cdot 2^{n+3}+1)$. It contains the following classes:

1. (1)

$q_{0}q_{j}q_{j+1}\cdot h_{0}$, 2. (2)

$q_{0}^{2}q_{j+1}\cdot h_{j}$, 3. (3)

$q_{0}^{2}q_{j}\cdot h_{j+1}$, 4. (4)

$q_{j-1}^{2}q_{j+1}\cdot h_{1}$, 5. (5)

$q_{1}q_{j-1}q_{j+1}\cdot h_{j-1}$, 6. (6)

$q_{1}q_{j-1}^{2}\cdot h_{j+1}$, 7. (7)

$q_{j}^{3}\cdot h_{1}$, 8. (8)

$q_{1}q_{j}^{2}\cdot h_{j}$.

Using 4.10(1) we compute the relevant algAH $d_{1}$-differentials. These differentials are displayed in Figure 8. In particular, the algAH sseq is empty in degree $(s,k,t)=(1,3,3\cdot 2^{n+3}+1)$ starting from the $E_{2}$-page. ∎

Lemma 4.22**.**

$\operatorname{\mathrm{Ext}}^{1,\ 3\cdot 2^{n+3}}_{\mathcal{P}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathrm{Ext}}_{\mathcal{Q}}^{2}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2}))\cong 0$.

Proof.

Dividing by $q_{0}$ as in 4.14 again we can determine the $E_{1}$-page of the algAH sseq in degree $(s,k,t)=(1,2,3\cdot 2^{n+3})$. It contains the following classes:

1. (1)

$q_{j}q_{j+1}\cdot h_{0}$, 2. (2)

$q_{0}q_{j+1}\cdot h_{j}$, 3. (3)

$q_{0}q_{j}\cdot h_{j+1}$,

Using 4.10(1) we compute the relevant algAH $d_{1}$-differentials. These differentials are displayed in Figure 9. In particular, the algAH sseq is empty in degree $(s,k,t)=(1,2,3\cdot 2^{n+3})$ starting from the $E_{2}$-page. ∎

Proof (of 4.2)..

The various components of this proposition appeared in the preceding lemmas in this subsection. ∎

5. The Cartan–Eilenberg spectral sequence

In this section we show that the Cartan–Eilenberg sseq has no differentials in a neighborhood around $h_{j}^{3}$ and use this to complete the proof of 3.2.

Lemma 5.1**.**

Let $j\geq 5$. There are no CE differentials entering the following $(a,t-a)$ degrees111Recall that $a=s+k$. $(4,3\cdot 2^{j}-4)$, $(5,3\cdot 2^{j}-4)$, $(6,3\cdot 2^{j}-4)$ or $(7,3\cdot 2^{j}-4)$.

Proof.

Recall that Cartan-Eilenberg $d_{r}$-differentials changes the tri-degrees in the following way

[TABLE]

Rewriting this in the $(a,s,t-a)$ basis we obtain

[TABLE]

From the sparsity of the CE sseq we know that all differentials have odd length. The following table contains a list of each differential we must rule out, and why each differential doesn’t occur.

[TABLE]

In each case 4.2 provides the required information about the source group. ∎

Recollection 5.2**.**

Building on Lin’s work on the 4-line, in [Che11, Theorem 1.2], Chen gives a complete description of $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}$ up to Adams filtration 5. In particular, Chen’s work provides the following information about $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}$ in a neighborhood of $h_{j}^{3}$:

[TABLE]

for $j\geq 8$. For $j=7$ degree $(a,t-s)=(5,3\cdot 2^{7}-3)$ contains the additional class $h_{8}D_{3}(1)$. For $j=6$ degree $(a,t-s)=(5,3\cdot 2^{6}-4)$ contains the additional class $h_{7}D_{3}(0)$.

Proposition 5.3**.**

Let $j\geq 5$. On the $E_{2}$-page of the classical Adams spectral sequence for $S^{0}$ the class $h_{0}^{3}g_{j-2}$ is non-zero.

Proof.

In the cases $j=5,6$, this is known from stemwise calculation of the $\operatorname{\mathrm{Ext}}$-groups. See for example [Bru, IWX20a] for $j=5$ and [Nas] for $j=6$. We will prove the proposition for $j\geq 7$.

The class $q_{0}^{3}$ survives the algAH sseq and CE sseq, and detects the class $h_{0}^{3}$ in the Adams $E_{2}$-page. From the Frobenius isomorphism between $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{*,2*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$ and $\operatorname{\mathrm{Ext}}_{\mathcal{A}}^{*,*}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$, the element $g_{n}$ in $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{4,3\cdot 2^{n+3}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$ detects $g_{n+1}$ in $\operatorname{\mathrm{Ext}}_{\mathcal{A}}^{4,3\cdot 2^{n+3}}(\operatorname{\mathbb{F}}_{2},\operatorname{\mathbb{F}}_{2})$. Therefore, the element $q_{0}^{3}\cdot g_{n}$ detects the element $h_{0}^{3}g_{n+1}$ in the Adams $E_{2}$-page.

For this we observe that $h_{0}^{3}g_{j-2}$ is detected on the Cartan–Eilenberg $E_{2}$-page by $q_{0}^{3}g_{j-3}$ and that this class is not hit by a CE differential since there are no differential which enter this tridegree by 5.1. ∎

Together 5.2 and 5.3 complete the proof of 3.2(1).

Lemma 5.4**.**

Let $j\geq 6$. The $E_{2}$-page of the Cartan–Eilenberg spectral sequence takes the following form near $h_{j}^{3}$:

[TABLE]

For $j=7$ there is an additional class, $h_{8}D_{3}(1)$, in degree $(a,s,t-a)=(5,5,3\cdot 2^{7}-3)$. For $j=6$ there is possibly an additional class, $h_{7}D_{3}(0)$, in degree $(a,s,t-a)=(5,4,3\cdot 2^{6}-4)$.

Proof.

Using the Frobenius isomorphism $\operatorname{\mathrm{Ext}}_{\mathcal{P}}^{s,2t}\cong\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{s,t}$ the information on lines 1, 3 and 6 can be extracted from Lin’s computation of $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}$ through Adams filtration 5 (see [Che11, Theorem 1.2]).

Recall that we proved that the CE $E_{2}$-page contains a non-trivial class $h_{0}g_{j-2}$ in 4.2. Suppose the CE $E_{2}$-page had another element $x$ besides $h_{0}g_{j-2}$ in degree $(a,s,t-a)=(5,4,3\cdot 2^{j}-4)$. This element would be a permanent cycle for degree reasons and would survive to the $E_{\infty}$-page since no differentials enter this tridegree by 5.1. This would imply that the rank of degree $(a,t-a)=(5,3\cdot 2^{j}-4)$ in the Adams $E_{2}$-page is at least $2$. This contradicts Lin’s calculations of $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}$, therefore no such $x$ can exist and we obtain the conclusion in line 2. In the $j=6$ case there the rank of $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}$ is 2 so we may have another class detecting $h_{7}D_{3}(0)$.

Recall that 4.2 lets us reduce proving the claims on lines 4 and 5 to showing that $q_{0}^{2}h_{j}^{3}\neq 0$. From Lin’s computations we know that $h_{0}^{2}h_{j+1}^{3}\neq 0$ on the Adams $E_{2}$-page. $h_{0}^{2}h_{j+1}^{3}$ would be detected on the CE $E_{2}$-page by $q_{0}^{2}h_{j}^{3}$ if this class was non-zero and if this class is zero, then it must be detected in $(a,s,t-a)=(5,5,3\cdot 2^{j}-3)$. On the other hand in $(a,s,t-a)=(5,5,3\cdot 2^{j}-3)$ the only class is $h_{1}g_{j-2}$ which is already detecting $h_{1}g_{j-2}$ (in the $j=7$ case there is also $h_{8}D_{3}(1)$, but this is already detecting $h_{8}D_{3}(1)$). The conclusion follows. ∎

Together 4.2 and 5.4 complete the proof of 3.2(3). We end the section by using the motivic CE sseq (and 2.10 in particular) to prove 3.2(2,4).

Lemma 5.5**.**

Let $j\geq 6$. There are no CE differentials entering the following $(a,t-a)$ degrees $(3,3\cdot 2^{j}-3)$, $(4,3\cdot 2^{j}-3)$ and $(5,3\cdot 2^{j}-3)$.

Proof.

Convergence of the CE sseq implies that the rank of the CE $E_{2}$-page is an upper bound on the rank of the Adams $E_{2}$-page and that this bound is sharp in degree $(a,t-a)$ exactly when there are no differentials entering or leaving degree $(a,t)$. Comparing the ranks from 3.2(3) with the ranks from 3.2(1) we see the bound is sharp and obtain the desired conclusion. ∎

Proof (of 3.2(2,4))..

Using 2.10 we combine the statements about Cartan–Eilenberg differentials from Lemmas 5.1 and 5.5 with our knowledge of the Cartan–Eilenberg $E_{2}$-page to prove 3.2(2).

In order to prove 3.2(4) we begin by observing that 2.10 lets us match up names of classes with names of lifts, so it suffices to determine where each class goes under Betti realization. In $(a,t-a)$ degrees $(4,3\cdot 2^{j}-4)$, $(5,3\cdot 2^{j}-4)$, $(3,3\cdot 2^{j}-3)$ and $(4,3\cdot 2^{j}-3)$ the target of Betti realization is a single $\operatorname{\mathbb{F}}_{2}$ so the conclusion is automatic. In $(a,t-a)$ degrees $(6,3\cdot 2^{j}-4)$, $(7,3\cdot 2^{j}-4)$ and $(5,3\cdot 2^{j}-3)$ we can choose our lifts of Cartan–Eilenberg permanent cycles to be the products $h_{0}^{2}\cdot g_{j-2}$, $h_{0}^{3}\cdot g_{j-2}$, $h_{0}\cdot h_{j}^{3}$, $h_{0}^{2}\cdot h_{j}^{3}$ and $h_{1}\cdot g_{j-2}$ and the compatibility of Betti realization with products lets us conclude. ∎

6. The key algebraic Novikov differential

In this section we prove the key motivic Adams differentials for $S^{0,0}/\tau$ appearing in 3.4. Through [GWX21, Theorem 1.17] this is equivalent to the following family of algebraic Novikov differentials:

Proposition 6.1**.**

In the algebraic Novikov spectral sequence we have differentials

[TABLE]

for $j\geq 5$.

Remark 6.2**.**

Note that in Proposition 6.1 we are using Cartan–Eilenberg names rather than motivic Adams names. In particular, the class $q_{0}^{3}g_{j-2}$ corresponds to the class $h_{0}^{3}g_{j-1}$ in 3.4 and the condition $j\geq 5$ corresponds to the condition $j\geq 6$.222See Section 5 for more on this translations of names.

Speaking broadly, the proof of 6.1 has two parts. In the first part, we relate the differential on $h_{j}^{3}$ to a product $\vartheta_{j}\vartheta_{j+1}$ on the Adams–Novikov $E_{2}$-page. Where $\vartheta_{j}$ refers to a particular choice of class that maps to $h_{j}^{2}$ under the Thom reduction map

[TABLE]

whose precise definition will be given in 6.6. In the second part, we use explicit cocycle representatives to identify $\vartheta_{j}\vartheta_{j+1}$ in the algebraic Novikov spectral sequence and thereby prove 6.1.

Remark 6.3**.**

In Section 7, we will explain how the product $\vartheta_{j}\vartheta_{j+1}$ is related to the Adams differentials supported by the Kervaire invariant class $h_{j}^{2}$ and is therefore of independent interest.

Speaking concretely, the section is organized as follows. In Section 6.1 we explain how algebraic Novikov differentials can be computed in terms of cocycles and reduce 6.1 to understanding the product $\vartheta_{j}\vartheta_{j+1}$. In Section 6.2 we identify the class detecting the product $\vartheta_{j}\vartheta_{j+1}$ in the algebraic Novikov spectral sequence. In Section 6.3, we provide material which, although not strictly necessary for the proof of 6.1, is useful for the reader looking to extend our results and methods.

Remark 6.4**.**

The family of differentials we prove in 6.1 appears as though it might be obtainable inductively via algebraic Steenrod operations. Unfortunately, as the moduli of formal groups is an integral object, the operation we would like to use is not definable without a lift of Frobenius. This does, however, suggest an alternative approach to proving 6.1 for $j\gg 0$.

Instead of studying $\mathcal{M}_{\mathrm{fg}}$ we can study another closely related object,

[TABLE]

which does admit a lift of Frobenius. This is the Witt vectors of the perfection of the reduction mod $p$ of the moduli of formal groups. Although there is no comparison map between these two stacks, it seems that computations on the Witt vector side can be lifted back to the moduli of formal groups in an asymptotic sense (i.e. for $j\gg 0$). On the Witt vector side the lift of Frobenius would let us reduce the proof of the analog of 6.1 to a single calculation.

6.1. Algebraic Novikov differentials in terms of cocycles

In Section 2 we introduced the algebraic Novikov spectral sequence by filtering the cobar complex for $\mathrm{BP}_{*}\mathrm{BP}$ by powers of the augmentation ideal $I$ of $\mathrm{BP}_{*}$. This spectral sequence takes the form

[TABLE]

In this subsection we explain how algebraic Novikov differentials can be calculated using cocycles and prove 6.1 modulo a lemma which we defer to Section 6.2.

Notation 6.5**.**

We write $\mathrm{cb}(-)$ for the cobar complex of a Hopf algebroid and use bar notation for elements of this complex. For example this means that we write $[t_{1}|t_{2}^{2}]$ for the class $t_{1}\otimes t_{2}^{2}$ in $(\mathrm{BP}_{*}\mathrm{BP})\otimes_{\mathrm{BP}_{*}}(\mathrm{BP}_{*}\mathrm{BP})$. Note that we are also identifying $\mathrm{BP}_{*}\mathrm{BP}$ with $\mathrm{BP}_{*}[t_{1},t_{2},\dots]$ in the usual way.

We let $I\cdot\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$ denote the (levelwise) augmentation ideal of this cosimplicial ring. The quotient by this ideal is $\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP}/I)$.

Suppose we have a class $\epsilon$ on the $E_{2}$-page of the algebraic Novikov spectral sequence. For simplicity we will assume that the $k$-degree of $\epsilon$ is zero. The differentials $d_{*}(\epsilon)$ can be computed using the following procedure:

1. (1)

Pick a cocycle $e$ in $\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP}/I)$ which represents $\epsilon$. 2. (2)

Pick a lift $e_{1}$ of $e$ to the cobar complex $\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$. Note that although $e$ is a cocyle, $e_{1}$ may not be a cocycle.333This is the mechanism by which algebraic Novikov differentials arise. 3. (3)

Compute $d(e_{1})$ in $\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$ and denote it by $c_{1}$. Note that $c_{1}\in I\cdot\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$ since $e_{1}$ is a cocycle modulo the augmentation ideal. Now proceed to step $(4.1)$. 4. (4.r)

We currently have $e_{r}$ with $c_{r}=d(e_{r})$ such that $c_{r}\in I^{r}\cdot\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$. In order to proceed we break into two cases:

• •

Suppose there exists a $w_{r}\in I\cdot\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$ such that

[TABLE]

Pick such a $w_{r}$ and proceed to step $(4.r+1)$ with

[TABLE]

(Note that $c_{r+1}=d(e_{r+1})\in I^{r+1}\cdot\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$.) This corresponds to the statement that $d_{r}(\epsilon)=0$.444More specifically, the class $w_{r}$ can be viewed as a class which supports a differential pre-empting the one on $\epsilon$. Note that this implies that in order to compute the $d_{r}$-differential one needs knowledge of all shorter length differentials.

• •

If no such $w_{r}$ exists, then we may conclude that there is a class

[TABLE]

which is detected by $c_{r}$ and survives to the $E_{r}$-page of the algebraic Novikov spectral sequence where it is hit by the differential $d_{r}(\epsilon)=\zeta$.

In order to apply this procedure to the classes $h_{j}^{3}$ there is one more wrinkle we need to iron out. In the statement of 6.1 both the source and target of the differential we wish to prove were stated in terms of the names for classes coming from the Cartan–Eilenberg $E_{2}$-page. This means we will have to provide a precise procedure for translating between these naming schemes.

From Section 2 we recall that the isomorphism

[TABLE]

sends a $q_{i}$ to $v_{i}$ and uses the fact that $\mathrm{BP}_{*}\mathrm{BP}/I\cong\mathcal{P}$ in order to provide an isomorphism at the level of cobar complexes. The most subtle point here is that this isomorphism sends $t_{i}$ to $\chi(\xi_{i}^{2})$ where $\chi$ is the antipode on $\mathcal{P}$.

Definition 6.6**.**

Let $T_{j}$ denote the following class in the cobar complex $\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$:

[TABLE]

The class $T_{j}$ is a cocycle since $d^{2}=0$ and the $\mathrm{BP}_{*}\mathrm{BP}$-cobar complex is torsion free. We will let $\vartheta_{j}$ denote the class on the Adams–Novikov $E_{2}$-page detected by $T_{j}$. In 6.10 we will check that $\vartheta_{j}$ maps to $h_{j-1}^{2}$ under the quotient map to the cohomology of $\mathcal{P}$.

For 6.1 we will need the following lemma whose proof we defer to the next subsection.

Lemma 6.7**.**

For $j\geq 5$, the product $\vartheta_{j}\vartheta_{j+1}$ is detected by $q_{0}^{2}g_{j-2}$ in the algebraic Novikov spectral sequence.

Now we prove 6.1.

Proof of 6.1.

We follow the procedure outlined above. The first step is picking a lift of $h_{j}^{3}$ to the the cobar complex $\mathrm{cb}(\mathrm{BP}_{*}\mathrm{BP})$. We pick the class:

[TABLE]

where $T_{j+1}$ is the cocycle given in 6.6 which detects $\vartheta_{j+1}$. We can compute the algebraic Novikov differential on $h_{j}^{3}$ by applying the cobar differential to this cocycle. In particular, we have

[TABLE]

Now we need to produce the correction terms for step (4). From 6.7 we know for $j\geq 5$, $\vartheta_{j}\vartheta_{j+1}$ is detected by $q_{0}^{2}g_{j-2}$ in the algebraic Novikov spectral sequence. In terms of cocycles this means that there exists a correction term $c$ such that

[TABLE]

and the image of this cocycle in the cohomology of $I^{3}/I^{4}$ is detected by $q_{0}^{3}g_{j-2}$.

Therefore, we have

[TABLE]

which is detected by $q_{0}^{3}g_{j-2}$. This implies that for $j\geq 5$,

[TABLE]

∎

6.2. A key product relation

In this subsection we prove 6.7, identifying the class detecting the product $\vartheta_{j}\vartheta_{j+1}$ in the algebraic Novikov spectral sequence in the following refined form.

Lemma 6.8**.**

For $j\geq 5$, there exists a correction term $c_{j}$ such that $T_{j}T_{j+1}+d(c_{j})$ is zero modulo $(4,v_{1}^{4})$. Moreover, the image of this cocycle in $I^{2}/I^{3}$ is detected by $q_{0}^{2}g_{j-2}$.

Remark 6.9**.**

This lemma is sufficient to conclude that $q_{0}^{2}g_{j-2}$ is a permanent cycle in the algebraic Novikov spectral sequence, but does not guarantee that this class is non-zero (even on the $E_{2}$-page). For this we need the material from Section 5.

The correction term $c_{j}$ we use is

[TABLE]

We will return to the issue of how we produced this correction term in the next subsection and at that point the reason for our somewhat unnatural choice of line breaks will become clear. The remainder of the proof involves computing $T_{j}|T_{j+1}+d(c_{j})$. We will do this by showing that this cocycle stabilizes for $j\gg 0$ and then using a small computer program to verify the desired conclusion holds in a single (now universal) case directly.

We will need to give formulas for the cobar differential on certain powers of $t_{1}$ and $t_{2}$ modulo $(8,v_{1}^{4})$. The key observation here is the following congruence of binomial coefficients

[TABLE]

This congruence allows us to replace cocycles which, a priori, would have a number of terms depending on $j$ with cocycles that are independent of $j$ in a certain sense. As an example to illustrate the point:

[TABLE]

Example 6.10**.**

Using the mod 4 version of these congruences we can determine the image of $\vartheta_{j}$ under the quotient map to the cohomology of $\mathcal{P}$.

[TABLE]

Under the isomorphism between $\mathrm{BP}_{*}\mathrm{BP}/I$ and $\mathcal{P}$ followed by the isomorphism with $\operatorname{\mathcal{A}}$ the class $[t_{1}^{2^{j-1}}]$ is detected by $h_{j-1}$ and therefore the image of $\vartheta_{j}$ maps to $h_{j-1}^{2}$.

This same formula also allows us to rewrite $T_{j}|T_{j+1}$ modulo $8$ in terms of powers of $t_{1}$ for $j\geq 3$

[TABLE]

Some further formulas we will need are

[TABLE]

[TABLE]

[TABLE]

In each case we have indexed the given formula so that it is valid for each $n\geq 0$.

Lemma 6.11**.**

The value of $T_{j}|T_{j+1}+d(c_{j})$ modulo $(8,v_{1}^{4})$ stabilizes for $j\geq 5$ in the sense that after expanding this cocycle as a sum of monomials in $t_{1}$ and $t_{2}$ we can increase $j$ by taking a termwise square.

Proof.

We’ve already expanded $T_{j}|T_{j+1}$ using the formula for $\Delta(t_{1}^{2^{n+3}})$. From this we can read off that the conclusion of the lemma is valid for this part of the cocycle once $j\geq 3$. Examining the terms which appear in $c_{j}$ we see that they are all produced from products and bars of the following terms

[TABLE]

where the underlined terms only appear with coefficient 2. The formulas provided above then give us the desired stabilization once $j\geq 5$. ∎

Proof (of 6.8)..

Using either the program provided in Appendix A or pencil, paper and patience one may give an explicit expansion of the cocycle $T_{5}|T_{6}+d(c_{5})$ modulo $(8,v_{1}^{4})$. Applying 6.11 we can then covert this into an explicit expansion of $T_{j}|T_{j+1}+d(c_{j})$ valid for $j\geq 5$.555Note that if one does this expansion by hand it is actually easier to skip over 6.11 and just use the formulas given prior to that lemma directly. We reproduce this expansion below (with $n=j-3$ for notational compactness):

[TABLE]

Since every term is divisible by $4$ we learn that $T_{j}|T_{j+1}+d(c_{j})$ is zero modulo $(4,v_{1}^{4})$ as desired. In order to determine which class detects this cocycle in $I^{2}/I^{3}$ we first note that no $v_{n}$’s appear other than $v_{0}=2$. This means that if we divide this class by $4$, view it as an element of the cobar complex of $\mathcal{P}$, and at that point it detects a class $H$, then the original cocycle was detected by $q_{0}^{2}H$ in the algebraic Novikov spectral sequence.

In order to determine the relevant class in the cohomology of $\mathcal{P}$ we make two observations. First, as a consequence of the results of [Lin08] the only nontrivial element of the cohomology of $\mathcal{P}$ in this bidegree is $g_{j-2}$, therefore it suffices to show that this class is detected by something nontrivial. Second, in the May spectral sequence $g_{j-2}$ is detected by $h_{2,j-2}^{4}$ (see [Tan70]). In order to identify our class we would like to pass to the associated graded of the May filtration. However, the $t_{i}$’s don’t directly correspond to the $\xi_{i}$’s in $\mathcal{P}$—they differ by an application of the anti-involution. Thankfully, this map is the identity on the May $E_{0}$-page so we can safely ignore this issue. After filtering out by terms of May filtration 11 and lower we obtain

[TABLE]

Passing to the May $E_{1}$-page we can rewrite this as

[TABLE]

which is equal to $h_{2,j-2}^{4}$, finishing the proof. ∎

Corollary 6.12**.**

The product $\vartheta_{j}\vartheta_{j+1}$ is nontrivial on the Adams–Novikov $E_{2}$-page.

Proof.

From 6.7 we know that $\vartheta_{j}\vartheta_{j+1}$ is detected by $q_{0}^{2}g_{j-2}$ in the algebraic Novikov spectral sequence. This means that it will suffice to show this class isn’t hit by a differential.

Examining the information about the $E_{2}$ page of the algebraic Novikov spectral sequence from 3.2(3) the only potential differentials which could hit $q_{0}^{2}g_{j-2}$ are a $d_{2}$ differential on $q_{0}h_{j}^{3}$ or a $d_{3}$ differential on $h_{j}^{3}$. We showed that $d_{2}(h_{j}^{3})$ and $d_{3}(h_{j}^{3})$ are zero in the Proposition 6.1. ∎

6.3. Constructing the correction term

In this short subsection we digress and discuss the choice of the correction term $c_{j}$. Although $c_{j}$ was originally produced by inspection we offer the following as evidence that it is perhaps not so difficult to produce such a term. We hope that this serves as a useful guide to the reader trying to make similar calculations in the future.

First note that we have rigged things so that at each step no $v_{i}$ appears (other than $v_{0}=2$). What this means is that the associated graded of the algebraic Novikov filtration becomes, as far as we see, the same as the cobar complex for $\mathcal{P}$. What gain does this provide us? Well, we can now filter things by the May filtration and successively add terms to work our way up the May filtration. As an example: The cocycle we started with $T_{j}|T_{j+1}$ is quite complicated, but mod 2 things are manageable

[TABLE]

We use the term $[t_{1}^{2^{j-1}}|t_{1}^{3\cdot 2^{j-1}}|t_{1}^{2^{j}}]$ to swap the middle pair of powers of $t_{1}$. Next we add the term $[t_{2}^{2^{j-1}}|t_{1}^{2^{j-1}}|t_{1}^{2^{j}}]$ in order to use the May $d_{1}$ differential killing $h_{j-1}h_{j}$.

At this point, after adding $d(-)$ of these two correction terms we get something which is zero mod 2. This means we get to move up one stage in the algebraic Novikov filtration. Now we look at things mod 4, since we get a cocycle mod 4 which is divisible by 2 we consider it again as a cocycle in $\mathcal{P}$. We also have to make a choice of lift of our coefficients on the correction term and we use the ones that appear in $c_{j}$ (i.e. one and seven). The reasoning behind this choice is related to how we initially wrote down this cocycle (by inspection) and it likely makes little difference in the end what coefficients were chosen.

The next stage involves examining the cocycle

[TABLE]

The leading term in the May filtration of this cocycle is $2[t_{2}^{2^{j-2}}|t_{2}^{2^{j-2}}|t_{1}^{2^{j-1}}|t_{1}^{2^{j}}]$ in May filtration 8. In the May $E_{1}$ page this cocycle is detected by $h_{2,j-2}^{2}h_{1,j-1}h_{1,j}$ and this class is hit by a May $d_{1}$ differential coming off of $h_{2,j-2}^{2}h_{2,j-1}$. Using this we conclude that the next correction term we want to add is $2[t_{2}^{2^{j-2}}|t_{2}^{2^{j-2}}|t_{2}^{2^{j-1}}]$.

The leading May filtration of the cocycle

[TABLE]

is $2[t_{2}^{2^{j-2}}|t_{1}^{2^{j-2}}|t_{1}^{2^{j-1}}|t_{2}^{2^{j-1}}]+2[t_{1}^{2^{j-2}}|t_{1}^{2^{j-1}}|t_{2}^{2^{j-2}}|t_{2}^{2^{j-1}}]$ in May filtration 8. In the May $E_{1}$ page this cocycle is already zero so we add a correction term that swaps the $t_{1}$’s and $t_{2}$’s so that they cancel (see the third line of the formula for $c_{j}$). After adding this correction the top May filtration is now reduced from 8 to 6.

Repeating this process we eventually eliminate the entire thing—proving that $\vartheta_{j}\vartheta_{j+1}$ is divisible by 4 (with the desired correction term $c_{j}$ providing a witness to divisibility in the cobar complex). In Appendix A we have annotated the corrections terms we add with the May lengths of the corresponding May differentials we are using at each step. One of the reasons this process was relatively straightforward was that we used only May $d_{0}$ and $d_{1}$ differentials.

7. The inductive approach revisited

In this section, which can for the most part be read independently of the rest of the paper, we investigate the fate of the Kervaire invariant one classes in the Adams spectral sequence. The celebrated Hill–Hopkins–Ravenel theorem on Kervaire invariant one tells us that $h_{j}^{2}$ supports a non-trivial Adams differential for every $j\geq 7$ [HHR16]. However, their work provides no further identifying information about these differentials.666For example, it remains possible that the lengths of these differentials grows without bound as $j$ increases. As a corollary of our study of these classes we provide a new lower bound on the length of the HHR differentials, showing that $d_{4}(h_{j}^{2})=0$.

We obtain our lower bound by revisiting one of the most promising proposals for constructing $\Theta_{j}$, the inductive approach of Barratt–Jones–Mahowald [BJM83]. The crux of this approach is a construction which proceeds

[TABLE]

In light of the HHR theorem this approach must break down at some point, meaning at least one of $\Theta_{5}^{2}$ or $\Theta_{6}^{2}$ is non-zero. The idea we pursue in this section is that it should be possible to run the inductive approach internal to the Adams spectral sequence on a fixed page. The main result of this section, 7.16, can be informally summarized as saying that,

[TABLE]

The power of this result lies in the fact that the inductive hypothesis we must verify lies 2 pages prior to the conclusion. This means that if we proceed by induction on $r$ (rather than induction on $j$) the classes $\theta_{j}^{2}$ are already defined for all $j$ at the outset. This opens the possibility that $\theta_{j}^{2}$ might be identified on the Adams $E_{r}$-page simultaneously for all $j$ and that being non-zero we might, then read off the HHR differentials. In order to make the expected outcome as concrete and precise as possible we make the following conjecture, which is a special case of Conjecture 1.22.

Conjecture 7.1** (Uniform Kervaire differentials conjecture).**

There exists a $Sq^{0}$-family of classes $(\mathrm{HHR})_{j}$ on the Adams $E_{2}$-page (defined for $j\gg 0$) and another class $x$ on the $E_{2}$-page such that

[TABLE]

In order to make the idea of “working on a fixed page of the Adams spectral sequence” rigorous we will work in the category of $\operatorname{\mathbb{F}}_{2}$-synthetic spectra introduced in [Pst18]. In the first subsection of this section we provide a lightning introduction to this category and its connection to the Adams spectral sequence. In the second subsection we run the Barratt–Jones–Mahowald inductive argument internal to the category of $\operatorname{\mathbb{F}}_{2}$-synthetic spectra. In the final subsection we investigate a certain product related to the differential on $h_{j}^{3}$.

Remark 7.2**.**

It is in this section that the distinction between $\Theta_{j}$, $\theta_{j}$ and $h_{j}^{2}$ becomes important and for this reason we pause to remind the reader of our conventions. We use $\Theta_{j}$ for the Kervaire invariant one classes in the stable homotopy groups of spheres. We use $h_{j}^{2}$ for the class on the Adams $E_{2}$-page. We use $\theta_{j}$ for a choice of class in the synthetic homotopy groups of $\operatorname{\mathbb{S}}/\!\lambda^{k}$ (which exists when $h_{j}^{2}$ survives to the $E_{k+1}$-page of the Adams spectral sequence).

7.1. A lightning introduction to synthetic spectra

In this subsection we provide a minimal introduction to the category of $\operatorname{\mathbb{F}}_{2}$-synthetic spectra. For a more complete introduction focusing on the construction of this category see [Pst18]. For a short introduction tailored to using synthetic spectra for computational purposes see [BHS19, Sections 9 and A]. For a more comprehensive account of synthetic homotopy theory see the (forthcoming) book [Bur23]. Since our main application is only in the case of $\operatorname{\mathbb{F}}_{2}$ we will stick to this case throughout our exposition.

7.1.1. Synthetic spectra

Construction 7.3** (Pstrągowski).**

There is a stable, presentably symmetric monoidal $\infty$-category $\mathrm{Syn}_{\operatorname{\mathbb{F}}_{2}}$ which fits into a diagram of symmetric monoidal functors

{\operatorname{\mathrm{Sp}}}$${\mathrm{Syn}_{\operatorname{\mathbb{F}}_{2}}}$${\operatorname{\mathrm{Sp}}}$${\mathrm{Stable}_{\operatorname{\mathcal{A}}}}$$\scriptstyle{1}$$\scriptstyle{\nu}$$\scriptstyle{(\operatorname{\mathbb{F}}_{2})_{*}(-)}$$\scriptstyle{\lambda^{-1}}$$\scriptstyle{-\otimes\operatorname{\mathbb{S}}/\!\lambda}

with the following properties:

1. (1)

$\nu$ commutes with filtered colimits and sends a cofiber sequence to a cofiber sequence if and only if it induces a short exact sequence on $\operatorname{\mathbb{F}}_{2}$-homology.777This condition on cofiber sequences is exactly the minimal one so that we have a long exact sequence at the level of the Adams $E_{2}$-page. 2. (2)

The functors $-\otimes\operatorname{\mathbb{S}}/\!\lambda$ and $\lambda^{-1}$ each commute with colimits. 3. (3)

The functor $\lambda^{-1}$ is a localization.

For each spectrum $X$ the object $\nu X$ records all information present in the Adams spectral sequence for $X$. In fact, one can reasonably think about the object $\nu X$ as being the Adams spectral sequence for $X$.

7.1.2. Synthetic homotopy groups

Before we can justify our claim that $\nu X$ records the data present in the Adams spectral sequence for $X$ we will need to introduce a way to measure a synthetic spectrum. For this our preferred method is via the bigraded synthetic homotopy groups and the action of the canonical bigraded homotopy element $\lambda$ on them. For the reader familiar with motivic spectra over $\mathbb{C}$ this pattern should be relatively familiar.

Definition 7.4** ([Pst18, Definitions 4.6 and 4.9]).**

The bigraded synthetic sphere $\operatorname{\mathbb{S}}^{k,s}$ is defined888We warn the reader that our conventions differ from those in some previous references. With the conventions used here the two indices of $\operatorname{\mathbb{S}}^{k,s}$ are the $x$ and $y$ coordinates in an Adams chart. We have found that in practice these are the most user-friendly conventions. to be $\Sigma^{-s}\nu S^{k+s}$. As is standard we will omit the superscripts in the case $(0,0)$, using the symbol $\operatorname{\mathbb{S}}$ for the monoidal unit of $\mathrm{Syn}_{\operatorname{\mathbb{F}}_{2}}$. For any synthetic spectrum $X$, the bigraded homotopy group $\pi_{k,s}(X)$ is defined to be the abelian group of homotopy classes of maps with source $\operatorname{\mathbb{S}}^{k,s}$ and target $X$.

Definition 7.5** ([Pst18, Definition 4.27]).**

For each spectrum $X$ we have an assembly map $\Sigma\nu X\to\nu\Sigma X$. In the case of $S^{-1}$ this provides us with a canonical map999Typically this class is referred to as $\tau$. Since we make use of both the synthetic and motivic categories, we denote it by $\lambda$ instead.

[TABLE]

The class $\lambda\in\pi_{0,-1}\operatorname{\mathbb{S}}$ and its behavior is the most important feature of the category of synthetic spectra. As an example say that $X$ is $\lambda$-invertible if the map

[TABLE]

is an equivalence. Then, the symmetric monoidal localization functor associated to the map $\lambda$ is the functor $\lambda^{-1}$ given above. In particular, the full subcategory of $\lambda$-invertible objects is equivalent to the category of spectra. At the opposite extreme, let the symbol $\operatorname{\mathbb{S}}/\!\lambda$ denotes the cofiber of $\lambda$, then we have a simple description of the homotopy groups of $\operatorname{\mathbb{S}}/\!\lambda\otimes X$.

Lemma 7.6** ([Pst18, Lemma 4.56]).**

For any spectrum $X$, there is a natural isomorphism of bigraded abelian groups

[TABLE]

between homotopy groups mod $\lambda$ and the $E_{2}$-page of the Adams spectral sequence for $X$.

In fact, this lemma is only a prelude to [BHS19, Theorem 9.19] which gives a precise translation101010Again we warn the reader that our grading differs by a shearing from those used in loc. cit. That said, $k$ and $s$ are the same. between Adams spectral sequence data for $X$ and bigraded synthetic homotopy groups for $\nu X$. Again for the readers familiar with motivic spectra over $\mathbb{C}$ this pattern should be relatively familiar.

7.1.3. The $\lambda$-Bockstein tower

With the class $\lambda$ in hand we can refine our understanding of the diagram in 7.3.

Theorem 7.7** (Pstrągowski).**

1. (1)

The cofiber of $\lambda$, which we denote $\operatorname{\mathbb{S}}/\!\lambda$, admits the structure of a commutative algebra in $\mathrm{Syn}_{\operatorname{\mathbb{F}}_{p}}$. 2. (2)

There is a natural symmetric monoidal equivalence

[TABLE]

from the stable $\infty$-category of modules over $\operatorname{\mathbb{S}}/\!\lambda$ to Hovey’s stable $\infty$-category of comodules over the Steenrod algebra $\operatorname{\mathcal{A}}$. The composite of $\nu$ with $\operatorname{\mathbb{S}}/\!\lambda\otimes-$ and this equivalence is naturally equivalent to the $\operatorname{\mathbb{F}}_{p}$-homology functor.

In fact, we can go beyond just $\operatorname{\mathbb{S}}/\!\lambda$ and study $\operatorname{\mathbb{S}}/\!\lambda^{n}$ for varying $n$. To do this in a coherent way we will need [BHS20, Example C.15] and the surrounding material. As a consequence of this $\mathrm{Syn}_{\operatorname{\mathbb{F}}_{p}}$ has the structure of a locally filtered category in the sense of [Lur15] where the canonical shift map is $\lambda$. Through this we make the following construction.

Construction 7.8**.**

The locally filtered structure on $\mathrm{Syn}_{\operatorname{\mathbb{F}}_{p}}$ comes from a symmetric monoidal left adjoint

[TABLE]

which sends $S(1)$ to $\operatorname{\mathbb{S}}^{0,1}$. This provides us with a tower of commutative algebras

[TABLE]

Using $r_{n,m}$ to denote these restriction maps, we have cofiber sequences

[TABLE]

The commutative algebra structure on $\operatorname{\mathbb{S}}/\!\lambda^{n}$ provides us with a symmetric monoidal category of modules over this object. In view of the connection between the $\lambda$-Bockstein and Adams differentials we think of the category of $\operatorname{\mathbb{S}}/\!\lambda^{n}$-modules as providing a way to work at the level of the Adams $E_{n+1}$-page.

Warning 7.9**.**

Outside the $n=1$ case, the bigraded homotopy groups of $\operatorname{\mathbb{S}}/\!\lambda^{n}$ are not literally given by the Adams $E_{n+1}$-page. Instead, thinking through the mechanics of the $\lambda$-bockstein spectral sequence one finds that there is a surjective map

[TABLE]

with kernel generated by the image of $\underline{\lambda}$ together with the $\lambda^{n-1}$-torsion classes.

7.1.4. Examples of synthetic homotopy groups

In order to prepare for later subsections we work through several examples of synthetic homotopy groups. Each of our examples will correspond to understanding a small region of the Adams spectral sequence for the sphere near $h_{j}^{2}$. Before proceeding, we suggest the reader new to synthetic spectra look at [BHS19, Section A.2] which works through an example chosen for its instructiveness in full detail.

{sseqdata}

[ name = thetaASS, xscale=1.25, yscale=1.25, x range = -50, y range = 05, x tick step = 2, y tick step = 2, class labels = left, classes = fill, grid = crossword, Adams grading, lax degree]

\class

"h_j^2" \class(-2,3) \structline\class(-2,4) \structline\class(-2,5) \structline\class(-2,6) \structline\class(-2,7) \structline

\class

(-1,3) \structline(-2,2) \class(0,4) \structline\class(1,5) \structline

\class

"h_j+1" \class(-1,2) \structline\class(-1,3) \structline\class(-1,4) \structline\class(-1,5) \structline\class(-1,6) \structline

\class

(0,2) \structline(-1,1) \class(1,3) \structline

[̣red]2(-1,1)(-2,3) [̣red]2(-1,2)(-2,4) [̣red]2(-1,3,-1)(-2,5) [̣red]2(-1,4)(-2,6) [̣red]2(-1,5)(-2,7)

{sseqdata}

[ name = theta7ASS, xscale=1.0, yscale=1.0, x range = 251256, y range = 05, x tick step = 2, y tick step = 2, class labels = left, classes = fill, grid = crossword, Adams grading, lax degree]

\class

"h_7^2" \class(256-2,3) \structline\class(256-2,4) \structline\class(256-2,5) \structline\class(256-2,6) \structline\class(256-2,7) \structline

\class

(256-1,3) \structline(256-2,2) \class(256,4) \structline\class(256+1,5) \structline

\class

"h_8" \class(256-1,2) \structline\class(256-1,3) \structline\class(256-1,4) \structline\class(256-1,5) \structline\class(256-1,6) \structline

\class

(256,2) \structline(256-1,1) \class(256+1,3) \structline

\class

"V_0’" below \class"K_1" below \class"D_3(2)" below \class(256,5) \structline

[̣red]2(256-1,1)(256-2,3) [̣red]2(256-1,2)(256-2,4) [̣red]2(256-1,3,-1)(256-2,5) [̣red]2(256-1,4)(256-2,6) [̣red]2(256-1,5)(256-2,7)

{sseqdata}

[ name = theta8ASS, xscale=1.0, yscale=1.0, x range = 507512, y range = 05, x tick step = 2, y tick step = 2, class labels = left, classes = fill, grid = crossword, Adams grading, lax degree] ı= 512

\class

"h_8^2" \class(512-2,3) \structline\class(512-2,4) \structline\class(512-2,5) \structline\class(512-2,6) \structline\class(512-2,7) \structline

\class

(512-1,3) \structline(512-2,2) \class(512,4) \structline\class(512+1,5) \structline

\class

"h_9" \class(512-1,2) \structline\class(512-1,3) \structline\class(512-1,4) \structline\class(512-1,5) \structline\class(512-1,6) \structline

\class

(512,2) \structline(512-1,1) \class(512+1,3) \structline

\class

"V_1’" below

[̣red]2(512-1,1)(512-2,3) [̣red]2(512-1,2)(512-2,4) [̣red]2(512-1,3,-1)(512-2,5) [̣red]2(512-1,4)(512-2,6) [̣red]2(512-1,5)(512-2,7)

{sseqdata}

[ name = theta6ASS, xscale=1.0, yscale=1.0, x range = 123127, y range = 07, x tick step = 2, y tick step = 2, class labels = left, classes = fill, grid = crossword, Adams grading, lax degree]

\class

"x_6,94" \class(124,7) \structline\class(124,6) \class(124,6)

\class

"D_3(1)"

\class

"K_0" \class(125,6) \structline\class(125,7) \structline\structline(124,6,1) \class"h_6H_1"

\class

(126,6) \class(126,7) \structline(125,6,-1)

\class

"h_6^2" \class(128-2,3) \structline\class(128-2,4) \structline\class(128-2,5) \structline\class(128-2,6) \structline\class(128-2,7) \structline\class(128-2,8) \structline\class(128-2,9) \structline

\class

(128-1,3) \structline(128-2,2) \class(128,4) \structline\class(128+1,5) \structline

\class

"h_7" \class(128-1,2) \structline\class(128-1,3) \structline\class(128-1,4) \structline\class(128-1,5) \structline\class(128-1,6) \structline\class(128-1,7) \structline\class(128-1,8) \structline

\class

(128,2) \structline(128-1,1) \class(128+1,3) \structline

\class

(127,6)

\class

(127,7) \structline(124,6) \class(127,7) \structline(126,6) \class(127,7) \class(127,7)

[̣red]2(128-1,1)(128-2,3,) [̣red]2(128-1,2)(128-2,4,-1) [̣red]2(128-1,3,-1)(128-2,5) [̣red]2(128-1,4)(128-2,6,-1) [̣red]2(128-1,5)(128-2,7,-1) [̣red]2(128-1,6)(128-2,8,-1) [̣red]2(128-1,7)(128-2,9,-1)

[̣red]2(125,5)(124,7)

In Figure 10 we display the Adams spectral sequence for the sphere near the class $h_{j}^{2}$ for111111In the $j=9$ case which is not pictured there is a single extra dot, $V_{2}^{\prime}$, located at $(-1,5)$. $j\geq 10$. The structure of the $E_{2}$-page in this range can be obtained from [Che11] and the indicated differentials are the Hopf invariant one differentials proved by Adams [Ada60]. The class $h_{1}h_{j+1}$ is a permanent cycle detecting the class $\eta_{j+1}$ in the homotopy groups of spheres constructed by Mahowald [Mah77].

Lemma 7.10**.**

Near $h_{j}^{2}$ for $j\geq 9$ the synthetic homotopy groups satisfy

[TABLE]

where $\theta_{j}$ is a choice of lift of $h_{j}^{2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{3}$.

In Figure 11 we display the Adams spectral sequence for the sphere around the classes $h_{7}^{2}$ and $h_{8}^{2}$. As above, the structure of the $E_{2}$-page is obtained from [Che11] and the indicated differentials are the Hopf invariant one differentials. Additionally, we will show in 7.17 that $d_{3}(h_{8}^{2})=0$.

Lemma 7.11**.**

Near $h_{7}^{2}$ and $h_{8}^{2}$ the synthetic homotopy groups satisfy

[TABLE]

where $\theta_{7}$ is a choice of lift of $h_{7}^{2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{3}$

[TABLE]

and $\theta_{8}$ is a choice of lift of $h_{8}^{2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{3}$.

We leave it as an instructive exercise for the reader to work out the proofs of Lemmas 7.10, 7.11 using [BHS19, Theorem 9.19].

In Figure 12 we display the Adams spectral sequence for the sphere near the class $h_{6}^{2}$. In this case our knowledge of the $E_{2}$-page comes from [Bru] and the $d_{2}$ differentials were computed in [Chu21b] using an implementation of an algorithm of Nassau refining Baues’ work on algorithmic computation of Adams $d_{2}$ differentials. Applying [BHS19, Theorem 9.19] we obtain the following lemma.

Lemma 7.12**.**

Near $h_{6}^{2}$ the synthetic homotopy groups satisfy

[TABLE]

where $\theta_{6}$ is a choice of lift of $h_{6}^{2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{3}$, $[D_{3}(1)]$ is a choice of lift of $D_{3}(1)$ to $\operatorname{\mathbb{S}}/\!\lambda^{2}$.

Remark 7.13**.**

The reader may have noticed that, somewhat counter-intuitively, we have chosen to present our examples in the opposite of the usual order. Our reason for doing this is to emphasize that the complexity of $E_{2}$ page is minimal for the generic element in a $\operatorname{\mathrm{Sq}}^{0}$-family. We expect that this phenomenon is robust.

7.1.5. Power operations

The study of power operations in the Adams spectral sequence began with work of Adams, Barratt and Mahowald on the quadratic construction. These ideas were developed further by several others over the next two decades attaining a relatively stable form with Bruner’s treatment in [BMMS86]. In order to give our synthetic inductive argument we will need to introduce the synthetic incarnation of this material.121212For a more relaxed introduction see the corresponding chapter in [Bur23]

For the purposes of this paper we only need the simplest example of a power operation: the quadratic construction. This construction, which can be performed in any symmetric monoidal category, takes a map $f$ with target the unit and sends it to the composite

[TABLE]

where $\mu$ is the multiplication map provided by the symmetric monoidal structure.

The quadratic construction uses three things: colimits, the symmetric monoidal structure and the commutative algebra structure on the unit. Since a symmetric monoidal left adjoint preserves each of these we learn that the quadratic construction is compatible with such functors. Now consider the following span of symmetric monoidal left adjoints.

{\mathrm{Syn}_{\operatorname{\mathbb{F}}_{p}}}$${\mathrm{Mod}(\mathrm{Syn}_{\operatorname{\mathbb{F}}_{p}};\operatorname{\mathbb{S}}/\!\lambda^{n})}$${\operatorname{\mathrm{Sp}}}$${\mathrm{Stable}(\operatorname{\mathcal{A}})}$$\scriptstyle{\operatorname{\mathbb{S}}/\!\lambda^{n}\otimes-}$$\scriptstyle{\lambda^{-1}}$$\scriptstyle{\operatorname{\mathbb{S}}/\!\lambda\otimes_{\operatorname{\mathbb{S}}/\!\lambda^{n}}-}

Although not immediately obvious, the existence of this span essentially encodes (nearly) all known compatibilites between power operations and the Adams spectral sequence. As a demonstration of this we consider the example of a map of synthetic spheres.

Example 7.14**.**

Suppose we have a map $\alpha:\operatorname{\mathbb{S}}^{k,s}\to\operatorname{\mathbb{S}}^{0,0}$. Applying the quadratic construction we obtain an associated map

[TABLE]

The usual filtration on the $C_{2}$ orbits provides us with a filtration on the source whose associated graded is given by $\operatorname{\mathbb{S}}^{2k+i,2s-i}$ in its $i^{\text{th}}$ piece. The bottom piece of this filtration is a copy of $\operatorname{\mathbb{S}}^{2k,2s}$ and composing with the associated inclusion gives $\alpha^{2}$. Inverting $\lambda$ on $(\operatorname{\mathbb{S}}^{k,s})^{\otimes 2}_{hC_{2}}$ gives us $(S^{k})^{\otimes 2}_{hC_{2}}\simeq\Sigma^{k}\mathbb{R}\mathrm{P}^{\infty}_{k}$ which allows us to work out the attaching maps.131313Since all the attaching maps decrease $s$ they are uniquely determined by what happens on inverting $\lambda$, i.e. by the attaching maps in a stunted projective space. If we think about $\alpha$ in terms of the class it represents on inverting $\lambda$, then the (changing) values of the $s$-degree of the cells of the quadratic construction are recording lower bounds on the Adams filtrations of power operations.

Now let’s tensor down to $\operatorname{\mathbb{S}}/\!\lambda$. The map $\alpha$ now becomes some class $a$ on the Adams $E_{2}$-page. The induced filtration on the quadratic construction now splits because there are no attaching maps of the appropriate degree on the $E_{2}$-page, so

[TABLE]

This provides a family of power operations $Q_{i}$ for $i\geq 0$ where $Q_{0}(a)=a^{2}$. Examining [May70] we learn that $Q_{i}(a)=\operatorname{\mathrm{Sq}}^{s-i}(a)$ and $Q_{i}(a)=0$ for $i>s$. This tells us that the quadratic construction in the synthetic category unifies the algebraic Steenrod operations on the Adams $E_{2}$-page with the quadratic construction in the category of spectra.

Bruner’s formulas for Adams differentials on algebraic Steenrod squares can now be read off by examining the $\lambda$-Bockstein spectral sequence for $(\operatorname{\mathbb{S}}^{k,s})^{\otimes 2}_{hC_{2}}$ and pushing these differentials forward using $\operatorname{\mathrm{Sq}}(\alpha)$.

In general, if we think about $X$ in terms of its bigraded cells, then $\pi_{**}(\nu\operatorname{\mathbb{F}}_{p}/\lambda\otimes X)$ is a bigraded vector space which records the locations of these cells. Then using the fact that $\nu\operatorname{\mathbb{F}}_{p}/\lambda\otimes-$ is symmetric monoidal we can work out the cells needed for $X^{\otimes 2}_{hC_{2}}$.

Example 7.15**.**

Suppose that $X$ has two cells $e_{1}$ and $e_{2}$ where $e_{j}$ is in degree $(k_{j},s_{j})$. Then, $X^{\otimes 2}_{hC_{2}}$ has cells $Q_{i}(e_{j})$ for $i\geq 0$ in degree $(2k_{j}+i,2s_{j}-i)$ and a single cell $e_{1}e_{2}$ in degree $(k_{1}+k_{2},s_{1}+s_{2})$.

7.2. The inductive approach

We now come to the main task of this section: giving a synthetic refinement of the inductive approach to constructing $\Theta_{j}$. In order to orient the reader we give a brief outline of the main points of the original version of this argument.

1. (1)

Using the hypothesis that $2\Theta_{j}=0$ construct a map

[TABLE]

which is $\Theta_{j}$ on the bottom cell. 2. (2)

Apply the quadratic construction to $\widehat{\Theta}_{j}$. A cell diagram of the source of the resulting map in shown below.

{{Q_{3}(a)}}$${Q_{1}(b)}$${{Q_{2}(a)}}$${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{0}(b)}}$${{Q_{1}(a)}}$${{ab}}$${{Q_{0}(a)}}$$\scriptstyle{\eta}$$\scriptstyle{\eta}$$\scriptstyle{2}$$\scriptstyle{\eta}$$\scriptstyle{2}$$\scriptstyle{\eta}$$\scriptstyle{2}

Each cell in this diagram is labelled by its associated Dyer–Lashof operation where $a$ is the bottom cell of the copy of $\mathbb{S}^{0}/2$ and $b$ is the top cell. 3. (3)

Composing $D_{2}(\widehat{\Theta}_{j})$ with the inclusion of the bottom cell gives $\Theta_{j}^{2}$ and assuming we have a nullhomotopy of this map, the induced map out of the cell indicated in blue will be $\Theta_{j+1}$ and the difference of the pair of cells above it will record a nullhomotopy of $2\Theta_{j+1}$.

Before proceeding further we remark that since inverting $\lambda$ is symmetric monoidal and commutes with colimits any lift of this procedure to the synthetic category (using maps between spheres) will produce something lying over $\Theta_{j}$. In essence, all that this would do is record an Adams filtration bound on $\Theta_{j+1}$. The next theorem shows that if we linearize with respect to $\operatorname{\mathbb{S}}/\!\lambda^{r}$, then the same argument works though with a weaker output and an inductive hypothesis which is easier to check (a substantial victory)!

Theorem 7.16**.**

*Fix an $r\geq 2$ and suppose that $\theta_{j}$ is a lift of $h_{j}^{2}$ from $\operatorname{\mathbb{S}}/\!\lambda$ to $\operatorname{\mathbb{S}}/\!\lambda^{r}$. If $2\theta_{j}=0$ and $\lambda^{2}\theta_{j}^{2}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{r})$, then there exists a class $\theta_{j+1}$ lifting $h_{j+1}^{2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{r}$ such that $2\theta_{j+1}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{r})$. *

Throughout the proof of 7.16 we will work in the symmetric monoidal category of $\operatorname{\mathbb{S}}/\!\lambda^{r}$-modules. In order to simplify notation we will use $\mathbbm{1}$ to denote $\operatorname{\mathbb{S}}/\!\lambda^{r}$ (since this is the monoidal unit of the category) and $\mathbbm{1}^{k,s}$ to denote $\Sigma^{k,s}\mathbbm{1}$. Before proceeding we pause to point out the key ideas in this proof:

• •

Although we work in $\operatorname{\mathbb{S}}/\!\lambda^{r}$-modules, we can often maintain control over maps at the level of their image under the functor $\operatorname{\mathbb{S}}/\!\lambda\otimes_{\operatorname{\mathbb{S}}/\!\lambda^{r}}-$.

• •

In $\operatorname{\mathbb{S}}/\!\lambda$-modules we have good control over the quadratic construction since it induces the action of algebraic Steenrod squares on the cohomology of the Steenrod algebra.

Proof.

Using a choice of nullhomotopy of $2\theta_{j}$ we can write down a map

[TABLE]

which is $\theta_{j}$ on the bottom cell. For later use we record the following property (which all choices of $\widehat{\theta}_{j}$ share):

• ($*$)

Upon tensoring down to $\operatorname{\mathbb{S}}/\!\lambda$ the source of $\widehat{\theta}_{j}$ splits into two cells. The bottom cell maps to $\mathbbm{1}$ by $h_{j}^{2}$ while the top cell maps to $\mathbbm{1}$ by $h_{j+1}$ (as long as $r\geq 2$ and $j\geq 3$).

To verify ($*$) consider the map of $\lambda$-Bockstein spectral spectral sequences induced by $\widehat{\theta}_{j}$. Label the bottom cell of the source by $a$ and the top cell by $b$. Then, we have $\lambda$-Bockstein differential $d_{1}(b)=h_{0}a$. By naturality we learn $d_{1}(\widehat{\theta}_{j}(b))=h_{0}h_{j}^{2}\neq 0$, which implies $\widehat{\theta}_{j}(b)$ is the unique non-zero class in $\operatorname{\mathbb{S}}/\!\lambda$ in this degree—$h_{j+1}$141414Note that in this argument we have proved the Hopf invariant differentials rather than citing them. The reader should compare this argument with the one which can be made using Bruner’s power operations formulas..

Applying the quadratic construction (i.e. the functor $(-)^{\otimes 2}_{hC_{2}}$) and postcomposing with the multiplication map on $\mathbbm{1}$ we obtain a map

[TABLE]

which is $\theta_{j}^{2}$ on the bottom cell. Now we make a couple of observations. First, we can read off from 7.14 that the cell diagram for $(\operatorname{\mathbb{S}}^{2^{j+1}-2}/2)^{\otimes 2}_{hC_{2}}$ given above is also a cell structure for $(\mathbbm{1}^{2^{j+1}-2,2}/2)^{\otimes 2}_{hC_{2}}$ where the bigradings of the cells have constant weight (the difference of the two coordinates). Second, after tensoring down to $\operatorname{\mathbb{S}}/\!\lambda$ the source splits as a sum of (shifts of) copies of the unit and the Dyer–Lashof operations the cells were labelled by now correspond to the Dyer-Lashof operations (i.e. Steenrod square) they record in $\operatorname{\mathcal{A}}$-comodules.

As above, we will label the bottom and top cell of $\mathbbm{1}^{2^{j+1}-2,2}/2$ as $a$ and $b$ respectively. Using our identification of what $b$ does on tensoring down to $\operatorname{\mathbb{S}}/\!\lambda$ from ($*$) we can read off that the cell $Q_{0}(b)$ in $(\mathbbm{1}^{2^{j+1}-2,2}/2)^{\otimes 2}_{hC_{2}}$ will map to $\mathbbm{1}$ via $h_{j+1}^{2}$. This means that if we can remove the cells below this one, then we will have a construction of $\theta_{j+1}$. After including a subcomplex of $(\mathbbm{1}^{2^{j+1}-2,2}/2)^{\otimes 2}_{hC_{2}}$ we obtain a map

{{Q_{1}(b)-Q_{3}(a)}}$${{Q_{0}(b)}}$${{Q_{0}(a)}}$${{\mathbbm{1}}.}$$\scriptstyle{\lambda\widetilde{2}}$$\scriptstyle{\lambda^{2}\eta}$$\scriptstyle{\theta_{j}^{2}}

Note the powers of $\lambda$ which appear in the attaching maps of this cell structure due to the bigradings of the cells. Next we compose this map with the map that pushes that the $\lambda^{2}$ off the bottom as indicated,

{{\mathbbm{1}^{2^{j+2}-1,1}}}$${{Q_{1}(b)-Q_{3}(a)}}$${{\mathbbm{1}^{2^{j+2}-2,2}}}$${{Q_{0}(b)}}$${{\mathbbm{1}^{2^{j+2}-4,2}}}$${{Q_{0}(a)}}$${{\mathbbm{1}}.}$$\scriptstyle{1}$$\scriptstyle{\lambda\widetilde{2}}$$\scriptstyle{\lambda\widetilde{2}}$$\scriptstyle{1}$$\scriptstyle{\eta}$$\scriptstyle{\lambda^{2}\eta}$$\scriptstyle{\lambda^{2}}$$\scriptstyle{\theta_{j}^{2}}

Using our assumption that $\lambda^{2}\theta_{j}^{2}$ is nullhomotopic we can factor this composite through $\mathbbm{1}^{2^{j+2}-2,2}/2$ via a map which is $h_{j}^{2}$ on the bottom cell if we tensor down to $\operatorname{\mathbb{S}}/\!\lambda$. This map is a good choice of $\widehat{\theta}_{j+1}$ and its construction completes the proof. ∎

Remark 7.17**.**

The condition that $\lambda^{2}\theta_{j}^{2}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{r})$ is implied by asking that $\theta_{j}^{2}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{r-2})$. This means that as a corollary of the fact that $h_{j}^{4}=0$ for $j\geq 1$ we learn that $h_{j}^{2}$ lifts to $\operatorname{\mathbb{S}}/\!\lambda^{3}$. In particular, $d_{2}(h_{j}^{2})=0$ and $d_{3}(h_{j}^{2})=0$.

Proposition 7.18**.**

There exist lifts $\theta_{j}$ of the classes $h_{j}^{2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{4}$ with the property that $2\theta_{j}=0$ and $\lambda^{2}\theta_{j}^{2}=0$.151515As this is the farthest we lift $h_{j}^{2}$ in this paper we will hereafter fix our notation so that $\theta_{j}$ refers to the classes constructed in this proposition. In more classical language the first claim is equivalent to saying that $d_{4}(h_{j}^{2})=0$.

Proof.

We proceed by induction on $j$ using 7.16. The inductive step requires that we check at each step that $\lambda^{2}\theta_{j}^{2}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{4})$. In fact we will check the stronger statement that $\theta_{j}^{2}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{2})$.

For $j\leq 4$ this follows from the fact that $\Theta_{j}^{2}=0$. If $j\neq 7$, then the conclusion follows from the fact that $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{5,2^{j+2}+1}=0$ (see Figures 10, 11 and 12).

In the case $j=7$ we must rule out the possibility that $\theta_{6}^{2}=\underline{\lambda}V_{0}^{\prime}$ in $\pi_{252,4}(\operatorname{\mathbb{S}}/\!\lambda^{2})$. In 7.22, in the next subsection, we will give a Toda bracket expression $\theta_{6}\in\langle 2,\theta_{5},\lambda\theta_{5},\widetilde{2}\rangle$ in $\pi_{126,2}(\operatorname{\mathbb{S}}/\!\lambda^{2})$. Multiplying by $\theta_{6}$ and shuffling we have

[TABLE]

To proceed we would like to determine the value of the 3-fold $\langle\theta_{6},2,\theta_{5}\rangle\in\pi_{189,3}(\operatorname{\mathbb{S}}/\!\lambda^{2})$. From 3.2(1) we can read off that $\pi_{189,3}(\operatorname{\mathbb{S}}/\!\lambda^{2})\cong\operatorname{\mathbb{Z}}/4\{[h_{6}^{3}]\}$. Meanwhile, the synthetic analog of Moss’ theorem (see [Bur23]) allows us to conclude that the image of $\langle\theta_{6},2,\theta_{5}\rangle$ in $\pi_{189,3}(\operatorname{\mathbb{S}}/\!\lambda)$ is zero. As a consequence we have

[TABLE]

for some $a\in\{0,1\}$. Now, with some further shuffling we may conclude that

[TABLE]

where the final step uses 7.11 which tells us $\pi_{252,3}\operatorname{\mathbb{S}}/\!\lambda^{2}=0$. ∎

Corollary 7.19**.**

For $j\geq 4$ we have $\delta_{4,1}(h_{j+1})=\widetilde{2}\theta_{j}$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{3})$.

Proof.

Examining the map induced by $\widehat{\theta}_{j}$ on $\lambda$-Bockteins in view of ($*$) provides the desired conclusion. ∎

In the case of $h_{6}^{2}$ where we have a spherical class $\Theta_{5}$ to work with we can provide a more refined statement.

Proposition 7.20**.**

The class $h_{6}^{2}$ survives to the $E_{r+3}$-page of the Adams spectral sequence if and only if $\eta\theta_{5}^{2}=0$ in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{r})$.

Proof.

We examine the bottom two cells in the final diagram in the proof of 7.16. The (total) $\lambda$-Bockstein in the source of this map records the attaching map so

[TABLE]

By naturality we now learn that

[TABLE]

This implies that $h_{6}^{2}$ is a permanent cycle if and only if $\lambda\eta\Theta_{5}^{2}=0$ and more specifically that the $\lambda$-power divisibility of this product records the length of the Adams differential on $h_{6}^{2}$.

∎

Using this proposition and our knowledge of the synthetic stable stems through topological degree $95$ can give an analysis of the class $\eta\Theta_{5}^{2}$ and a corresponding lower bound on the length of an Adams differential on $h_{6}^{2}$.

Theorem 7.21**.**

The class $\eta\theta_{5}^{2}$ is divisible by $\lambda^{6}$. In particular, through the previous proposition we learn that $h_{6}^{2}$ survives to the Adams $E_{9}$-page.

Proof.

Classically, by Corollary 1.10 in [WX17], we have a strictly defined 4-fold Toda bracket for $\Theta_{5}$:

[TABLE]

by Corollary 1.3 in [Xu16], we have $2\Theta_{5}=0$, therefore

[TABLE]

Working synthetically, we also have

[TABLE]

[TABLE]

by comparing with classical statements and using that these bidegrees are $\lambda$-torsion free. Therefore,

[TABLE]

In order to run the main argument that proves the theorem we now need six additional facts whose proofs we defer for the moment.

1. (1)

$\pi_{93,3}\operatorname{\mathbb{S}}\subseteq\lambda^{2}\cdot\pi_{93,5}\operatorname{\mathbb{S}}$. In particular, any element in $\langle\theta_{5},2,\theta_{4}\rangle\subseteq\pi_{93,3}\operatorname{\mathbb{S}}$ can be written in the form $\lambda^{2}x$, where $x\in\pi_{93,5}\operatorname{\mathbb{S}}$. 2. (2)

For any $x$ as in $(1)$, we have $x\cdot\theta_{4}=0$ in $\pi_{123,7}\operatorname{\mathbb{S}}$. 3. (3)

For any $x$ as in $(1)$, any element in $\langle x,\theta_{4},2\rangle\subseteq\pi_{124,6}\operatorname{\mathbb{S}}$ can be written in the form $\lambda^{2}z$, where $z\in\pi_{124,8}\operatorname{\mathbb{S}}$. 4. (4)

$\pi_{124,8}\operatorname{\mathbb{S}}\subseteq\lambda\cdot\pi_{124,9}\operatorname{\mathbb{S}}$. 5. (5)

$\eta\cdot\pi_{124,9}\operatorname{\mathbb{S}}\subseteq\lambda\cdot\pi_{125,11}\operatorname{\mathbb{S}}$. 6. (6)

$\eta\cdot\pi_{31,1}\operatorname{\mathbb{S}}\subseteq\lambda^{4}\cdot\pi_{32,6}\operatorname{\mathbb{S}}$.

Fact $(1)$ lets us rewrite $\theta_{5}^{2}\in\langle\langle\theta_{5},2,\theta_{4}\rangle,\theta_{4},2\rangle$ as

[TABLE]

Now, by $(2)$, the bracket $\langle x,\theta_{4},2\rangle$ is well-defined. Choose a $y$ in this bracket, then we have

[TABLE]

Since both $\theta_{5}^{2}$ and $\lambda^{2}\cdot y$ are contained in $\langle\lambda^{2}x,\theta_{4},2\rangle$, their difference belongs to its indeterminacy, which is $\lambda^{2}x\cdot\pi_{31,1}\operatorname{\mathbb{S}}+2\cdot\pi_{124,4}\operatorname{\mathbb{S}}$. Therefore, we have

[TABLE]

Using $(3)$ and $(4)$, we may write $y=\lambda^{2}\cdot z=\lambda^{2}\cdot\lambda\cdot w$ for some $z\in\pi_{124,8}\operatorname{\mathbb{S}}$ and $w\in\pi_{124,9}\operatorname{\mathbb{S}}$. Multiplying our expression for $\theta_{5}^{2}$ by $\eta$ we obtain

[TABLE]

By $(5)$, $\eta\cdot w$ is $\lambda$-divisible; by $(6)$, $\eta\cdot\pi_{31,1}$ is $\lambda^{4}$-divisible. Therefore, $\eta\theta_{5}^{2}$ is $\lambda^{6}$-divisible.

We now return to proving the six facts from above.

1. (1)

In the 93-stem of the classical Adams sseq both $h_{5}^{3}$ and $h_{0}h_{5}^{3}$ support non-trivial differentials (see [IWX20b]). Synthetically, this means any element in $\pi_{93,3}\operatorname{\mathbb{S}}$ is divisible by $\lambda^{2}$. 2. (2)

Classically, $\langle\Theta_{5},2,\Theta_{4}\rangle\cdot\Theta_{4}=\Theta_{5}\cdot\langle 2,\Theta_{4},\Theta_{4}\rangle\subseteq\Theta_{5}\cdot\pi_{61}=0$, where the last equality is due to $\pi_{61}S^{0}=0$ (see Theorem 1.9 in [WX17]). This means that synthetically $\lambda^{2}x\cdot\theta_{4}$ is $\lambda$-torsion.

From Figure 12, we know that on the classical Adams $E_{2}$-page

[TABLE]

[TABLE]

These facts means that synthetically $\pi_{123,7}\operatorname{\mathbb{S}}$ contains no non-trivial $\lambda$-torsion elements. Therefore, we must have $x\cdot\theta_{4}=0$ in $\pi_{123,7}\operatorname{\mathbb{S}}$. 3. (3)

It is clear from the classical Adams $E_{2}$ and $E_{3}$-pages in Figure 12 that the synthetic homotopy group

[TABLE]

where $a_{124,6}$, $a_{124,7}$, $b_{124,7}$ are the generators of the classical $E_{3}$-page in the corresponding degrees. We need to rule out these three elements in $\pi_{124,6}(\operatorname{\mathbb{S}}/\!\lambda^{2})$.

For $a_{124,6}$, note that $x$ is detected by $h_{1}g_{3}$ or it is further $\lambda$-divisible. Then it can be ruled out by the corresponding Massey product

[TABLE]

in the Adams $E_{3}$-page and Moss’s theorem (there are no crossing differentials).

For the other two elements, note that both $a_{124,7}$, $b_{124,7}$ on the $E_{3}$-page have nonzero $h_{2}$-multiples. Therefore, we only need to show that

[TABLE]

does not contain (a linear combination of) $\nu\cdot\underline{\lambda}a_{124,7}$ and $\nu\cdot\underline{\lambda}b_{124,7}$.

In fact, since $h_{1}g_{3}\cdot h_{2}=0$ in $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{6,96+6}$, and $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{7,96+7}=0$, we know that $\nu\cdot x$ is $\lambda^{2}$-divisible. Therefore, in $\pi_{127,7}(\operatorname{\mathbb{S}}/\!\lambda^{2})$,

[TABLE]

Factoring $2$ as $\lambda\widetilde{2}$ we may use the isomorphism between the Adams $E_{3}$-page and the image of multiplication by $\lambda$ on $\pi_{**}\operatorname{\mathbb{S}}/\!\lambda^{2}$ to analyze $2$-divisibility. In particular, we can read off that $\nu\cdot\underline{\lambda}a_{124,7}$ and $\nu\cdot\underline{\lambda}b_{124,7}$ are not 2-divisible in $\pi_{127,7}(\operatorname{\mathbb{S}}/\!\lambda^{2})$ from the fact that $h_{2}a_{124,7}$ and $h_{2}b_{124,7}$ are not $h_{0}$-divisible on the $E_{3}$-page. 4. (4)

This is clear from Figure 12 since the only nonzero element in $\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{8,124+8}$ supports a nonzero $d_{2}$-differential. Therefore, we have $\pi_{124,8}\subseteq\lambda\cdot\pi_{124,9}\operatorname{\mathbb{S}}$. 5. (5)

This is clear from Figure 12 since $h_{1}\cdot\operatorname{\mathrm{Ext}}_{\operatorname{\mathcal{A}}}^{9,124+9}=0$. Therefore, we have $\eta\cdot\pi_{124,9}\operatorname{\mathbb{S}}\subseteq\lambda\cdot\pi_{125,11}\operatorname{\mathbb{S}}$. 6. (6)

It is clear from the classical Adams spectral sequence that $\pi_{31,1}\operatorname{\mathbb{S}}$ is generated by $\lambda^{2}\eta\theta_{4}$ and $\lambda^{4}$-multiples. So we only need to show that $\eta\cdot\lambda^{2}\eta\theta_{4}$ is $\lambda^{4}$-divisible. This is true since it is zero. In fact, classically $\eta^{2}\Theta_{4}=0$, so synthetically $\eta^{2}\theta_{4}$ must be a $\lambda$-torsion or zero. For degree reasons, it must be zero. Therefore, we have $\eta\cdot\pi_{31,1}\operatorname{\mathbb{S}}\subseteq\lambda^{4}\cdot\pi_{32,6}\operatorname{\mathbb{S}}$.

This completes the proof. ∎

7.3. A (synthetic) product of $\theta_{j}$’s

Having constructed synthetic lifts $\theta_{j}$ in $\operatorname{\mathbb{S}}/\!\lambda^{4}$ we make a short investigation of products among them. More specifically, we show that there is a (non-trivial) product

[TABLE]

on the Adams $E_{4}$-page (i.e. in the homotopy of $\operatorname{\mathbb{S}}/\!\lambda^{3}$)161616We have passed from $\operatorname{\mathbb{S}}/\!\lambda^{4}$ to $\operatorname{\mathbb{S}}/\!\lambda^{3}$ here in order to avoid potential extra terms in the product.. Though investigating this product may seem unmotivated, as it will turn out, knowledge of this product is essentially equivalent to 1.11 less the assertion that the differential is non-trivial.

Lemma 7.22**.**

For $j\geq 5$ and not $7$ the $\operatorname{\mathbb{S}}/\!\lambda^{3}$-linear 4-fold $\langle\widetilde{2},\lambda\theta_{j},\theta_{j},2\rangle$ is defined, contains $\theta_{j+1}$ and has no indeterminacy (except in the $j=5$ case where $\underline{\lambda}^{2}D_{3}(1)$ may be in the indeterminacy).

Proof.

We start with showing that the 4-fold is defined. In 7.18 we showed that $2\theta_{j}=0$ and $\lambda^{2}\theta_{j}^{2}=0$ in $\operatorname{\mathbb{S}}/\!\lambda^{4}$. This means that the 3-folds $\langle\widetilde{2},\lambda\theta_{j},\theta_{j}\rangle$ and $\langle\lambda\theta_{j},\theta_{j},\widetilde{2}\rangle$ are defined. After tensoring down to $\operatorname{\mathbb{S}}/\!\lambda^{3}$ we observe that since $\pi_{2^{j+2}-3,3}(\operatorname{\mathbb{S}}/\!\lambda^{3})=\pi_{2^{j+2}-3,2}(\operatorname{\mathbb{S}}/\!\lambda^{3})=0$ as proved in 7.10 and 7.11 both 3-folds are automatically zero.

In order to unambiguously identify the value of this 4-fold we begin by observing that the reduction map $\pi_{2^{j+2}-2,2}(\operatorname{\mathbb{S}}/\!\lambda^{3})\to\pi_{2^{j+2}-2,2}(\operatorname{\mathbb{S}}/\!\lambda)$ is an isomorphism (surjective with kernel $\underline{\lambda}^{2}D_{3}(1)$ in the $j=5$ case). As a consequence it will suffice to understand the value of this bracket after reduction to $\operatorname{\mathbb{S}}/\!\lambda$. The desired conclusion can now be obtained using the synthetic version of Moss’ theorem for 4-folds [Bur23]. ∎

As it would be much more involved to argue that $\langle\widetilde{2},\lambda\theta_{7},\theta_{7}\rangle$ is zero in $\pi_{**}(\operatorname{\mathbb{S}}/\!\lambda^{3})$ we instead prove a slightly modified form of 7.22 in the $j=7$ case.

Lemma 7.23**.**

In $\operatorname{\mathbb{S}}/\!\lambda^{3}$ the matric 4-fold

[TABLE]

is defined, contains $\theta_{8}$ and has zero indeterminacy.

Proof.

The proof of this lemma is quite similar to previous one, so we will mostly indicate the necessary modification. Each of the 2-fold products are zero as before. The right 3-fold is then defined and lands in a zero group, therefore it suffices to check that the left 3-fold contains zero. From 7.11 we know that $\pi_{509,3}(\operatorname{\mathbb{S}}/\!\lambda^{3})\cong\operatorname{\mathbb{Z}}/2\{\underline{\lambda^{2}}V_{1}^{\prime}\}$ which implies that the left 3-fold is equal to its own indeterminacy and therefore it contains zero. In order to evaluate the value of this 4-fold we begin with some shuffling,

[TABLE]

Next we remove the pair of zero maps through $\Sigma^{1,-3}\operatorname{\mathbb{S}}/\!\lambda$ reducing to an ordinary 4-fold. Note that in doing this we must take into account the indeterminacy coming from composites which factor through $\Sigma^{1,-3}\operatorname{\mathbb{S}}/\!\lambda$.

[TABLE]

Using the fact $\operatorname{\mathbb{S}}/\!\lambda^{3}$ is a free $\operatorname{\mathbb{S}}/\!\lambda^{3}$-module the adjunction between $\mathbb{S}$-modules and $\operatorname{\mathbb{S}}/\!\lambda^{3}$-modules allows us to read off that for $f\in[\Sigma^{510,2}\operatorname{\mathbb{S}}/\!\lambda^{3},\Sigma^{1,-3}\operatorname{\mathbb{S}}/\!\lambda]_{\operatorname{\mathbb{S}}/\!\lambda^{3}}$ and $g\in[\Sigma^{1,-3}\operatorname{\mathbb{S}}/\!\lambda,\operatorname{\mathbb{S}}/\!\lambda^{3}]_{\operatorname{\mathbb{S}}/\!\lambda^{3}}$ the composite $g\circ f$ depends only on the underlying $\mathbb{S}$-linear composite of $f$ and $g$. From the cofiber sequence $\operatorname{\mathbb{S}}^{1,-3}\to\Sigma^{1,-3}\operatorname{\mathbb{S}}/\!\lambda\to\operatorname{\mathbb{S}}^{2,-5}$ and the fact that $\pi_{1,-3}\operatorname{\mathbb{S}}/\!\lambda^{3}=0$ and $\pi_{2,-5}\operatorname{\mathbb{S}}/\!\lambda^{3}=0$ we can read off that $[\Sigma^{1,-3}\operatorname{\mathbb{S}}/\!\lambda,\operatorname{\mathbb{S}}/\!\lambda^{3}]=0$. Thus, we may conclude that the composite $g\circ f$ is zero and our previous expression simplifies to

[TABLE]

At point this we can conclude as in the previous lemma by observing that the value of this bracket is uniquely identified after reduction to $\operatorname{\mathbb{S}}/\!\lambda$ and then applying the synthetic version of Moss’ theorem for 4-folds [Bur23]. ∎

Lemma 7.24**.**

In $\operatorname{\mathbb{S}}/\!\lambda^{3}$ the product $\theta_{j}\theta_{j+1}$ is divisible by $\widetilde{2}$ for $j\geq 5$.

Proof.

We begin by handling the case $j\neq 5,7$. From 7.11 and 7.10 we know that $\pi_{2^{j+2}-3,3}\operatorname{\mathbb{S}}/\!\lambda^{3}=0$. This implies that $\langle\theta_{j},2,\theta_{j}\rangle$ is strictly zero. As a consequence of this relation we are able to shuffle the expression for $\theta_{j}$ given in 7.22 to produce the desired divisibility171717Note that shuffling gives a one-to-one correspondence of values of brackets so indeterminacy issues do not arise in this argument.

[TABLE]

In the cases $j=5,7$ we need a different argument to conclude that $\langle\theta_{j},2,\theta_{j}\rangle$ contains zero. Since $2\theta_{j}=0$ and $\operatorname{\mathbb{S}}/\!\lambda^{3}$ is an $\operatorname{\mathbb{E}}_{\infty}$-algebra (see 7.8) this bracket contains the product $2\cdot Q_{1}(\theta_{j})$ where here we are referring to the spherical power operation $Q_{1}$ on even degree classes first identified by Toda [Tod62, p.27-28]. Since all elements of $\pi_{125,3}(\operatorname{\mathbb{S}}/\!\lambda^{3})$ and $\pi_{509,3}(\operatorname{\mathbb{S}}/\!\lambda^{3})$ are simple 2-torsion we know $2Q_{1}(\theta_{j})=0$ for $j=5,7$. This finishes the $j=5$ case.

In the case $j=7$ we follow the same basic strategy replacing 7.22 with 7.23. In order to shuffle the 4-fold in this case we need the pair of 3-folds

[TABLE]

to be zero.181818More precisely, we need that we can always pick the nullhomotopy of the product on the right to make the bracket zero. The first of these brackets is zero since it lands in a zero group and we have already checked the second contains zero. ∎

The restriction placed on the product of $\theta_{j}$ and $\theta_{j-1}$ by 7.24 allows us to obtain a corresponding restriction on the Adams differentials on $h_{j}^{3}$.

Proposition 7.25**.**

$\delta_{4,1}(h_{j}^{3})$* is divisible by $\widetilde{2}^{2}$ for $j\geq 6$. Consequently, for $j\geq 6$,*

1. (1)

$d_{2}(h_{j}^{3})=0$, 2. (2)

$d_{3}(h_{j}^{3})$* is either [math] or $h_{0}^{2}g_{j-2}$ and* 3. (3)

$d_{4}(h_{j}^{3})$* is either [math] or $h_{0}^{3}g_{j-2}$ (if defined).*

Proof.

From 7.8 we know that $\delta_{4,1}$ arises as the cofiber of the map of rings $r_{4,1}:\operatorname{\mathbb{S}}/\!\lambda^{4}\to\operatorname{\mathbb{S}}/\!\lambda$ and therefore that this map is $\operatorname{\mathbb{S}}/\!\lambda^{4}$-linear. Using that $\theta_{j}$ lifts to $\operatorname{\mathbb{S}}/\!\lambda^{4}$ (by 7.16) we may now perform the following manipulations

[TABLE]

where we have used 7.19 to identify $\delta_{4,1}(h_{j})$. Plugging the $\widetilde{2}$ divisibility of $\theta_{j}\theta_{j-1}$ from 7.24 into this equation completes the proof of the first claim.

Let $\delta_{4,1}(h_{j}^{3})=\widetilde{2}^{2}\cdot y$ with $y\in\pi_{3\cdot 2^{j}-4,3}\operatorname{\mathbb{S}}/\!\lambda^{3}$. Using our knowledge of the $E_{2}$-page from [Che11] (cf. 3.2(1)) together with 1.23 which lets us lift $g_{j-2}$ to $\operatorname{\mathbb{S}}/\!\lambda^{2}$ (we denote the lift by $[g_{j-2}]$) we can read off that $\pi_{3\cdot 2^{j}-4,3}\operatorname{\mathbb{S}}/\!\lambda^{3}\cong\operatorname{\mathbb{Z}}/4\{\underline{\lambda}[g_{j-2}]\}$.191919In the case $j=6$ there is an additional summand $\operatorname{\mathbb{Z}}/2\{\underline{\lambda^{2}}(h_{7}D_{3}(0))\}$, but this class is $\widetilde{2}$-torsion and so does not affect the value of $\widetilde{2}^{2}\cdot y$. It follows that $d_{2}(h_{j}^{3})=0$ and depending on the value of $y\in\operatorname{\mathbb{Z}}/4\{\underline{\lambda}[g_{j-2}]\}$ we fall into one of the following cases

• (a)

$y$ is a generator and $d_{3}(h_{j}^{3})=h_{0}^{2}g_{j-2}$,

• (b)

$y=2\underline{\lambda}[g_{j-2}]$, $d_{3}(h_{j}^{3})=0$ and $d_{4}(h_{j}^{3})=h_{0}^{3}g_{j-2}$.

• (c)

$y=0$, $d_{2}(h_{j}^{3})=0$, $d_{3}(h_{j}^{3})=0$ and $d_{4}(h_{j}^{3})=0$.

∎

Now we can reverse the flow of information. From 1.11 we know that

[TABLE]

which, when combined with Equation 2, implies:

Corollary 7.26**.**

In $\operatorname{\mathbb{S}}/\!\lambda^{3}$, there is a product relation, $\theta_{j}\theta_{j-1}=\underline{\lambda^{2}}h_{0}^{2}g_{j-2}\neq 0$ for $j\geq 6$.

The family of products in this corollary are one of the first infinite families of hidden products in the Adams spectral sequence. It seems likely that they bear a close relations to the hidden products $\theta_{j}^{2}$ which identify the HHR differentials.

Appendix A Code

In this appendix we provide a short script which was used in Section 6. This script runs within the computer algebra system Sage and evaluates the cobar differential in the Hopf algebroid $(\mathrm{BP}_{*},\mathrm{BP}_{*}\mathrm{BP})$ modulo $v_{1}$ and a power of $2$ on terms which include only $t_{1}$ and $t_{2}$. Ultimately, we know this script gives the correct answer in our case of interest since we have checked the outputs by hand.

• •

The ring $K$ is the base ring for our computation. We use $\operatorname{\mathbb{Z}}/8$ since that is what is relevant for us, but other choices work as well.

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The rings $A$ and $B$ represent the third and fourth layers of the cobar complex respectively.

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The modifiers $a,b,c$ or $w,x,y,z$ indicate which tensor factor a given variable comes from.

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The cobar complex is a cosimplicial ring and the various $di$’s are relevant face maps of this cosimplicial ring.

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The term $E$ corresponds to the cocycle $T_{j}|T_{j+1}$ from Section 6.

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The final line (which is expanded and printed on evaluation) is the cocycle $T_{j}|T_{j+1}+d(c_{j})$.

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We have annotated the various terms which make up $c_{j}$ with the length of the May differential they are involved in (see Section 6.3).

K = Zmod(8); A.<t1a, t1b, t1c, t2a, t2b, t2c> = K[]; B.<t1w, t1x, t1y, t1z, t2w, t2x, t2y, t2z> = K[];

d0 = A.hom([t1x, t1y, t1z, t2x, t2y, t2z], B); d1 = A.hom([t1w + t1x, t1y, t1z, t2w - t1w * t1x^2 + t2x, t2y, t2z], B); d2 = A.hom([t1w, t1x + t1y, t1z, t2w, t2x - t1x * t1y^2 + t2y, t2z], B); d3 = A.hom([t1w, t1x, t1y + t1z, t2w, t2x, t2y - t1y * t1z^2 + t2z], B); d4 = A.hom([t1w, t1x, t1y, t2w, t2x, t2y], B);

def cobar_diff(x): return d0(x) - d1(x) + d2(x) - d3(x) + d4(x);

def make_theta_left(k): return 4 * t1w^(1 * 2^(k)) * t1x^(7 * 2^(k)) + 6 * t1w^(2 * 2^(k)) * t1x^(6 * 2^(k)) + 4 * t1w^(3 * 2^(k)) * t1x^(5 * 2^(k)) + 3 * t1w^(4 * 2^(k)) * t1x^(4 * 2^(k)) + 4 * t1w^(5 * 2^(k)) * t1x^(3 * 2^(k)) + 6 * t1w^(6 * 2^(k)) * t1x^(2 * 2^(k)) + 4 * t1w^(7 * 2^(k)) * t1x^(1 * 2^(k));

def make_theta_right(k): return 4 * t1y^(1 * 2^(k)) * t1z^(7 * 2^(k)) + 6 * t1y^(2 * 2^(k)) * t1z^(6 * 2^(k)) + 4 * t1y^(3 * 2^(k)) * t1z^(5 * 2^(k)) + 3 * t1y^(4 * 2^(k)) * t1z^(4 * 2^(k)) + 4 * t1y^(5 * 2^(k)) * t1z^(3 * 2^(k)) + 6 * t1y^(6 * 2^(k)) * t1z^(2 * 2^(k)) + 4 * t1y^(7 * 2^(k)) * t1z^(1 * 2^(k));

The n which appears here is related to j by n = j - 3.

n = 2;

theta_left = make_theta_left(n); theta_right = make_theta_right(n+1);

E = theta_left * theta_right;

term1 = t1a^(4 * 2^n) * t1b^(12 * 2^n) * t1c^(8 * 2^n); term2 = 7 * t2a^(4 * 2^n) * t1b^(4 * 2^n) * t1c^(8 * 2^n); term3 = 2 * t2a^(2 * 2^n) * t2b^(2 * 2^n) * t2c^(4 * 2^n); term4 = 2 * t1a^(2 * 2^n) * t1b^(4 * 2^n) * t2c^(4 * 2^n) * (t2a^(2 * 2^n) + t2b^(2 * 2^n)); term5a = 2 * t1a^(4 * 2^n) * t1b^(4 * 2^n) * t1c^(4 * 2^n); term5b = 2 * t1a^(2 * 2^n) * t1b^(2 * 2^n) * t1c^(8 * 2^n); term5 = (term5a + term5b) * (t2a^(4 * 2^n) + t2b^(4 * 2^n) + t2c^(4 * 2^n)); term6 = 2 * t1a^(6 * 2^n) * t1b^(10 * 2^n) * t1c^(8 * 2^n); term7 = 2 * t1a^(8 * 2^n) * t1b^(12 * 2^n) * t1c^(4 * 2^n) + 2 * t1a^(12 * 2^n) * t1b^(8 * 2^n) * t1c^(4 * 2^n); term8 = 2 * t1a^(2 * 2^n) * t2b^(2 * 2^n) * t1c^(16 * 2^n) + 2 * t1a^(4 * 2^n) * t2b^(4 * 2^n) * t1c^(8 * 2^n);

correction = term1 + term2; # first row correction += term3; # may d1 correction += term4; # may d0 correction += term5; # may d0 correction += term6; # may d0 correction += term7; # may d0 correction += term8; # may d1 E + cobar_diff(correction);