On the $v_1$ periodicity of the Moore space

TL;DR

This paper advances the understanding of the $v_1$-periodic component in the Adams spectral sequence for Moore spaces by refining existing algebraic methods and proposing a new conjecture to verify Mahowald's long-standing conjecture.

Contribution

It improves upon Palmieri's approach by using the endomorphism ring of the Moore space, leading to a new conjecture related to Mahowald's conjecture.

Findings

Progress in verifying Mahowald's conjecture for Moore spaces.

Development of a new algebraic framework using $End(M)$.

Formulation of a new conjecture to connect with Mahowald's formulation.

Abstract

We present progress in trying to verify a long-standing conjecture by Mark Mahowald on the $v_{1}$-periodic component of the classical Adams spectral sequence for a Moore space $M$. The approach we follow was proposed by John Palmieri in his work on the stable category of $A$-comodules. We improve on Palmieri's work by working with the endomorphism ring of $M$ - $End(M)$ thus resolving some of the initial difficulties of his approach and formulating a conjecture of our own that would lead to Mahowald's formulation.

Equations

$$E_2(M;Q_1)=Ex{t}_{Q_{1\ast \ast }{Q}_1}(Q_{1\ast \ast },Q_{1\ast \ast }(M))={\mathbb{F}}_2[v_1^{\pm 1},h_{11},h_{21},\cdots ,h_{n1},\cdots ]$$

$$\begin{array}[]{cc}d_{3}(v_{1}^{2})=h_{11}^{3}\\ d_{3}(h_{n1})=v_{1}^{-2}h_{11}h_{21}h_{n-1,1}^{2}&\text{for }n\geq 3\end{array}$$

$$\begin{array}[]{cc}d_{3}(v_{1}^{2})=h_{11}^{3}\\ d_{3}(h_{n1})=v_{1}^{-2}h_{11}^{3}h_{n1}+v_{1}^{-2}h_{11}h_{21}h_{n-1,1}^{2}&\text{for }n\geq 3\end{array}$$

$$E_2(S;Q_1)=Ex{t}_{Q_{1\ast \ast }{Q}_1}(Q_{1\ast \ast },Q_{1\ast \ast })={\mathbb{F}}_2[v_1^{\pm 1},h_{10},h_{11},h_{21},\cdots ,h_{n1},\cdots ]$$

$$E_2(M;Q_1)\cong E_2(S;Q_1)/(h_{10})={\mathbb{F}}_2[v_1^{\pm 1},h_{11},h_{21},\cdots ,h_{n1},\cdots ]$$

$$M\wedge DM\wedge M\wedge DM$$

$$(x_{i}y_{j})(x_{k}y_{l})=\begin{cases}\begin{array}[]{cc}x_{i}y_{l}&\text{if }j+k=0\\ 0&\text{otherwise}\end{array}\end{cases}$$

$$\psi (\alpha )=\psi (x_0{y}_{-1})=\psi (x_0)\psi (y_{-1})=(1\otimes x_0)(1\otimes y_{-1})=1\otimes x_0{y}_{-1}=1\otimes \alpha$$

$$\psi (\gamma )$$

$$E_2(End(M);Q_1)$$

$$Ex{t}_{{\mathbb{F}}_2[{\xi }_1]/({\xi }_1^4)}({\mathbb{F}}_2,{\mathbb{F}}_2⟨1,\alpha ,\gamma ,\alpha \gamma ⟩)$$

$$E_2(End(M);Q_1)={\mathbb{F}}_2[v_1^{\pm 1},\alpha ,h_{11},h_{21},h_{31},...]/({\alpha }^2)$$

$$M\wedge DM\wedge M$$

$${\eta }_{\ast }:{\mathbb{F}}_2[v_1^{\pm 1}]\otimes {\mathbb{F}}_2[h_{10},h_{11},h_{21},h_{31},...]\to {\mathbb{F}}_2[v_1^{\pm 1}]\otimes {\mathbb{F}}_2[h_{11},h_{21},h_{31},...]\otimes {\mathbb{F}}_2⟨1,\alpha ⟩$$

$$E_2(M;Q_1)\cong E_2(End(M);Q_1)/(\alpha )={\mathbb{F}}_2[v_1^{\pm 1},h_{11},h_{21},\cdots ,h_{n1},\cdots ]$$

$$d_3^M(h_{21}\cdot v_1)=d_3(h_{21})\cdot v_1+h_{21}\cdot d_3^M(v_1)=v_1^{-1}{h}_{11}^3{h}_{21}\ne 0$$

$$\text{(Main) Conjecture part 1: }{d}_2(h_{n1})=v_1^{-1}\alpha h_{11}^2{h}_{n1}\text{ for }n\ge 3$$

$$E_2(End(M);Q_1)={\mathbb{F}}_2[x_1,x_2,...]\otimes {\mathbb{F}}_2[v_1^{\pm 1},h_{11},\alpha ]/({\alpha }^2)$$

$$E_3(End(M);Q_1)={\mathbb{F}}_2[x_1,x_2,...]\otimes {\mathbb{F}}_2[v_1^{\pm 2},h_{11},\alpha ,{\alpha }^{\prime }]/({\alpha }^2,\alpha h_{11}^2,\alpha {\alpha }^{\prime },{\alpha }^{\prime 2})$$

$$\text{(Main) Conjecture part 2: }{d}_3(x_n)=v_1^{-4}{h}_{11}{x}_1{x}_{n-1}^2\text{ for }n\ge 2$$

$$d_2(v_1^{-2}{h}_{11}{h}_{21}{h}_{n-1,1}^2)=d_2(h_{21})v_1^{-2}{h}_{11}{h}_{n-1,1}^2=v_1^{-3}\alpha h_{11}^3{h}_{21}{h}_{n-1,1}^2$$

$$d_3(v_1^2{h}_{n1})=d_3(v_1^2)h_{n1}+v_1^2{d}_3(h_{n1})=h_{11}^3{h}_{n1}+h_{11}^3{h}_{n1}=0$$

$$d_3(v_1^2{x}_n)=d_3(v_1^2)x_n+v_1^2{d}_3(x_n)=h_{11}^3{x}_n+h_{11}^3{x}_n+v_1^{-2}{h}_{11}{x}_1{x}_{n-1}^2=v_1^{-2}{h}_{11}{x}_1{x}_{n-1}^2$$

$$0=d_3^2(v_1^2{x}_n)=d_3(v_1^{-2}{h}_{11}{x}_1{x}_{n-1}^2)=d_3(v_1^{-2})h_{11}{x}_1{x}_{n-1}^2=v_1^{-4}{h}_{11}^4{x}_1{x}_{n-1}^2$$

$$d_3^M(h_{n1})=d_3(v_1^{-2}{x}_{n-1})\cdot v_1=v_1^{-2}{h}_{11}^3{h}_{n1}+v_1^{-2}{h}_{11}{h}_{21}{h}_{n-1,1}^2$$

$$v_1^{-1}Ex{t}_A^{s,t}({\mathbb{F}}_2,H_{\ast }(M))$$

$$v_1^{-1}Ex{t}_A^{s,t}({\mathbb{F}}_2,H_{\ast }(bo\wedge M))={\mathbb{F}}_2[v_1^{\pm 4}]\otimes {\mathbb{F}}_2(h_{11},v_1)/(h_{11}^3,v_1^2)$$

$$v_1^{-1}Ex{t}_A^{s,t}({\mathbb{F}}_2,H_{\ast }(bu\wedge M))={\mathbb{F}}_2[v_1^{\pm 1}]$$

$$0$$

$$(i)\,\,\,\begin{array}[]{cc}\ker(d_{3}^{n})=Z(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{h_{11}^{n}\}&\text{if }n=0,1\\ \ker(d_{3}^{n})/Z(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{h_{11}^{n}\}\cong\\ \cong B(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{v_{1}^{2}\}\otimes\{h_{11}^{n-2}\}&\text{if }n\geq 2\end{array}$$

Full text

1 Introduction

Stable homotopy groups of spheres (or generally any finite complex) have long been a subject of study in algebraic topology. Due to their immense complexity, one might try to understand them from the lens of the Adams spectral sequence. However, this task as well seems outside of the scope of what we can fully understand. Nevertheless, the chromatic point of view allows us to “break” the spectral sequence into chromatic pieces. This paper represents an attempt to understand one of those pieces - namely, the $v_{1}$-periodic component of the Adams spectral sequence of a Moore space i.e. $v_{1}^{-1}E_{2}(M;H)$. A conjecture, which seems quite likely given the available data, was formulated in [1] by Mark Mahowald almost 50 years ago. Still, of course, no amount of data can be a substitute for a rigorous proof.

Palmieri proposed an approach to this conjecture in [2]. He built a generalized Adams spectral sequence in the stable category of comodules over the dual Steenrod algebra $A$. This spectral sequence converges to $v_{1}^{-1}E_{2}(M;H)$ and computations seem promising due to the simplicity of $E_{2}=\mathbb{F}_{2}[v_{1}^{\pm 1},h_{11},h_{21},\cdots,h_{n1},\cdots]$ and the fact that $E_{3}=E_{2}$ as for degree reasons nontrivial differentials can only occur at odd pages. It is important to note that since $M$ is not a ring spectrum, $E_{r}$ is not an algebra and $d_{r}$ is not a derivation and so what we really mean by the above equality is that $E_{2}$ is a $\mathbb{F}_{2}$-vector space with basis the monomials in $\mathbb{F}_{2}[v_{1}^{\pm 1},h_{11},h_{21},\cdots,h_{n1},\cdots]$. Palmieri then conjectured what the values of $d_{3}(h_{n1})$ are and proposed one should be able to extend them in some way to the entire $E_{3}$. Moreover he conjectured that the spectral sequence collapses at $E_{4}$ and claimed this would imply Mahowald’s conjecture. Note it is not immediately obvious how Palmieri’s formulation relates to Mahowald’s and it is something we address in more detail at a later section of the paper.

Thus our problem is three-fold: how does one compute $d_{3}(h_{n1})$, how does one extend it to the rest of $E_{3}$ and why are there no higher degree differentials. We solely address the first two questions, fully answering the second one. We do this by working with the endomorphism ring spectrum of $M$ - $End(M)$. It is the $4$ cell complex $M\wedge DM$. The advantage of $End(M)$ is that its spectral sequence is multiplicative and so $d_{3}$ is a derivation. At the same time the action $End(M)\wedge M\to M$ makes $E_{r}(M)$ into a module over $E_{r}(End(M))$. We will also show Palmieri’s originally conjectured values for $d_{3}(h_{n1})$ can’t be true and so we propose a revised conjecture of what those values are. We verify that conjecture modulo knowing that the elements $v_{1}^{m}h_{n1}$ don’t survive to $E_{4}$ for $n\geq 3$, $m\in\mathbb{Z}$.

In section $2$ we provide the necessary background about $Stable(A)$

• the stable category of comodules over the Steenrod algebra $A$, and explicitly write Palmieri’s original conjecture and our revised version of it. In section 3 we work out the corresponding spectral sequence for $End(M)$ and its action on the the one for $M$. Section $4$ consists of the meat of the paper as we proceed to prove our main results stated above. We conclude with section $5$ where we introduce the original conjecture by Mahowlad and show explicitly how it follows from our revised conjecture.

The author would also like to thank his advisor Haynes Miller for the unending support and multitude of fruitful discussions and suggestions, including the crucial idea of working with $End(M)$.

2 The category $Stable(A)$

In this chapter we give a brief description of $Stable(A)$ and any related results of immediate use to us. For more detail the reader is directed to Palmieri’s book [2].

Objects in $Stable(A)$ are unbounded cochain complexes of (left) $A$-comodules. We will identify a comodule $L$ with its injective resolution over $A$. For two such objects $L,N$ the set of morphisms is $[L,N]_{s,t}=Ext_{A}^{s,t}(L,N)$. Then $L_{s,t}=\pi_{s,t}(L)=Ext_{A}^{s,t}(\mathbb{F}_{2},L)$. For the sake of clarity we observe $L$ itself is bigraded and one should make a distinction between the elements of degree $(s,t)$ in $L$ and $L_{s,t}$. Note also the sphere spectrum $S\in Stable(A)$ is the injective resolution of $\mathbb{F}_{2}$, which is in line with our notation of $\pi_{s,t}(L)=[S,L]_{s,t}$ above. $Stable(A)$ is now a triangulated category and for a ring spectrum $X\in Stable(A)$ we can build a generalaized Adams spectral sequence in the usual way. Then assuming certain conditions hold we can identify $E_{2}(L;X)=Ext_{X_{**}X}(X_{**},X_{**}L)$ and further conditions would guarantee convergence to $\pi_{**}L$.

We are interested in the case where the spectrum $Q_{1}$ plays the role of $X$. To define $Q_{1}$, we first define $q_{1}$ to be the injective resolution of $A\Box_{\mathbb{F}_{2}(\xi_{2})/(\xi_{2}^{2})}\mathbb{F}_{2}$. $Q_{1}$ is now obtained from $q_{1}$ after working out how to extend the $q_{1}$-resolution into the negative dimensions. Then one can check $q_{1**}=\mathbb{F}_{2}[v_{1}]$, $Q_{1**}=\mathbb{F}_{2}[v_{1}^{\pm 1}]$ [2, p.44] and $Q_{1**}Q_{1}=\mathbb{F}_{2}[v_{1}^{\pm 1},\xi_{1},\xi_{2}^{2},\cdots\xi_{n}^{2},\cdots]/(\xi_{1}^{4},\xi_{2}^{4},\cdots)$ [2, p.101].

The trigraded spectral sequence of interest is

[TABLE]

and it converges to $v_{1}^{-1}E_{2}(M;H)$ [2, p.81, 101]. Note the abuse of notation above as what we really mean by $E_{2}(M;Q_{1})$ is $E_{2}(L;Q_{1})$ where $L$ is an injective resolution for $H_{*}(M)$. Elsewhere $M$ will always refer to the topological Moore spectrum. For degree reasons the only potential non-zero differentials in $E_{r}(M;Q_{1})$ happen at odd pages, so $E_{2}=E_{3}$. Palmieri then conjectured the following differentials:

[TABLE]

As we will see later, the conjecture in its current form is incorrect, so we make the following revised conjecture:

[TABLE]

Though this isn’t enough to fully determine $d_{3}$, Palmieri goes on to propose that $d_{3}$ “looks” as though as $E_{2}(M;Q_{1})$ is an algebra. One reason for this proposal that he notes is we can also compute the $E_{2}$ page of the corresponding spectral sequence for the sphere

[TABLE]

and use the map $S\to M$ to induce a surjection $E_{2}(S;Q_{1})\to E_{2}(M;Q_{1})$ with $h_{n1}\to h_{n1}$, $h_{10}\to 0$ and $v_{1}\to v_{1}$. Then the identity map $S\wedge M\to M$ turns $E_{2}(M;Q_{1})$ into a cyclic module over $E_{2}(S;Q_{1})$. Now identifying $E_{2}(M;Q_{1})$ with $\mathbb{F}_{2}[v_{1}^{\pm 1},h_{11},h_{21},\cdots,h_{n1},\cdots]$ becomes justified as both coincide as $E_{2}(S;Q_{1})$-modules:

[TABLE]

Then information about differentials in $E_{r}(S;Q_{1})$ could directly produce differentials in $E_{r}(M;Q_{1})$ and since $S$ is a ring spectrum, $E_{r}(S;Q_{1})$ is a spectral sequence of algebras, so the differentials in $E_{r}(S;Q_{1})$ are derivations. The problem is differentials in $E_{2}(S;Q_{1})$ are difficult to compute and so we don’t know what $E_{3}(S;Q_{1})$ looks like. This is where $End(M)$ enters the picture - it is a ring spectrum that acts on $M$ just as $S$ does, but differentials in $E_{2}(End(M);Q_{1})$ are much more manageable to compute.

3 The $Q_{1}$ $E_{2}$ term for $End(M)$

We begin by computing $H_{*}(End(M))$ as a comodule over $A$. Let $x_{0}$ and $x_{1}$ denote the two cells of $M$ and $y_{-1}$ and $y_{0}$ denote the two cells of $DM=\Sigma^{-1}M$. Then $End(M)=M\wedge DM$ has four cells of the form $x_{i}y_{j}$ with $|x_{i}y_{j}|=i+j$. As $DM$ is the dual of $M$ we have maps $\eta:S\to M\wedge DM$ and $\epsilon:DM\wedge M\to S$ that specify the ring structure of $End(M)$. More precisely, $\eta$ is the unit, while multiplication is given by

[TABLE]

and the action of $End(M)$ on $M$ is then given by the map $1\wedge\epsilon:M\wedge DM\wedge M\to M$. If $\iota\in H_{*}(S)$ is the generator, then $\eta_{*}(\iota)=x_{1}y_{-1}+x_{0}y_{0}$ and $\epsilon_{*}(y_{1}x_{-1})=\epsilon_{*}(y_{0}x_{0})=\iota$. This allows us to compute the multiplicative structure of $H_{*}(End(M))$

[TABLE]

Setting $\alpha=x_{0}y_{-1}$ and $\gamma=x_{1}y_{0}$ we get that $H_{*}(End(M))=\mathbb{F}_{2}[\alpha,\gamma]/(\alpha^{2},\gamma^{2},\alpha\gamma+\gamma\alpha+1)$. Note this is a 4-dimensional non-commutative $\mathbb{F}_{2}$-algebra with basis $\langle 1,\alpha,\gamma,\alpha\gamma\rangle$ where $|\alpha|=-1$ and $|\gamma|=1$. To understand the coaction of $A$ we just need to understand the coaction on $\alpha$ and $\gamma$. Since $\psi(x_{0})=1\otimes x_{0}$ and $\psi(x_{1})=1\otimes x_{1}+\xi_{1}\otimes x_{0}$ we conclude that

[TABLE]

and

[TABLE]

Recall we are interested in computing $d_{3}$ in $E_{2}(M;Q_{1})$. Since $M$ lacks multiplicative structure, we will work with $End(M)$ and try to understand $E_{r}(End(M);Q_{1})$. We proceed with a direct computation

[TABLE]

Here we used that the coaction of $\xi_{i}^{2}$ on $\mathbb{F}_{2}\langle 1,\alpha,\gamma,\alpha\gamma\rangle$ is trivial for $i\geq 2$. The conormal extension $\mathbb{F}_{2}(\xi_{1}^{2})/(\xi_{1}^{4})\to\mathbb{F}_{2}(\xi_{1})/(\xi_{1}^{4})\to\mathbb{F}_{2}(\xi_{1})/(\xi_{1}^{2})$ produces a Cartan-Eilenberg spectral sequence that collapses since $H_{*}(End(M))=\mathbb{F}_{2}\langle 1,\alpha,\gamma,\alpha\gamma\rangle$ is cofree over $\mathbb{F}_{2}(\xi_{1})/(\xi_{1}^{2})$. Thus, we get

[TABLE]

We conclude that $Ext_{\mathbb{F}_{2}[\xi_{1}]/(\xi_{1}^{4})}(\mathbb{F}_{2},\mathbb{F}_{2}\langle 1,\alpha,\gamma,\alpha\gamma\rangle)=\mathbb{F}_{2}\langle 1,\alpha\rangle\otimes\mathbb{F}_{2}[h_{11}]$ and so

[TABLE]

which (expectedly so) is two copies of $E_{2}(M;Q_{1})$. The degrees of the generators are given by $|v_{1}|=(0,2,1),\,|\alpha|=(0,-1,0),\,|h_{n1}|=(1,2^{n+1}-2,0)$. It is worth noting that even though $H_{*}(End(M))$ is not commutative, the spectral sequence above ends up with a commutative multiplicative structure.

3.1 $E_{2}(M;Q_{1})$ as a differential module over $E_{2}(End(M);Q_{1})$

The action of $End(M)$ on $M$ extends to an action $E_{r}(End(M);Q_{1})\otimes E_{r}(M;Q_{1})\to E_{r}(M;Q_{1})$ and so $E_{r}(M;Q_{1})$ is a differential module over $E_{r}(End(M);Q_{1})$. The commutative diagram

[TABLE]

implies the action of $E_{2}(S;Q_{1})$ on $E_{2}(M;Q_{1})$ factors through the action of $E_{2}(End(M);Q_{1})$ via the algebra map $\eta_{*}:E_{r}(S,Q_{1})\to E_{r}(End(M);Q_{1})$, which is just

[TABLE]

with $\eta_{*}(v_{1})=v_{1}$ and $\eta_{*}(h_{n1})=h_{n1}$. Furthermore we claim $\eta_{*}(h_{10})=\alpha h_{11}$. Indeed, since $\psi(\gamma)=1\otimes\gamma+\xi_{1}\otimes 1+\xi_{1}^{2}\otimes\alpha$ it follows that $\xi_{1}\otimes 1+\xi_{1}^{2}\otimes\alpha$ vanishes in the homology of the cobar complex of $End(M)$ and so $\alpha h_{11}=\xi_{1}^{2}|\alpha=\xi_{1}|1$, which is the cobar representative of $h_{10}$ in $E_{2}(S;Q_{1})$.

Hence $E_{2}(M;Q_{1})$ is a cyclic module over $E_{2}(End(M);Q_{1})$. Furthermore, we have an isomorphism of $E_{2}(End(M);Q_{1})$-modules:

[TABLE]

Before we move on to the next section we note that all of the elements $h_{11},v_{1},h_{21}v_{1},h_{21}v_{1}^{2}$ survive to $E_{\infty}(M;Q_{1})$ as shown by the diagram of $E_{2}(M;H)$ below. Observe this doesn’t guarantee the same is true in $E_{r}(End(M);Q_{1}),$ but we will still be able to extract some of the information back to $E_{r}(End(M);Q_{1})$ using the action above.

4 Calculating $d_{2}$ and $d_{3}$ of $E_{2}(End(M);Q_{1})$

We begin by calculating $d_{2}$ and $d_{3}$ on the low-degree elements in $E_{r}(End(M);Q_{1})$ and then proceed to formulating a conjecture for $d_{2}$ and $d_{3}$ on the remaining elements.

Theorem 1: The elements $\alpha,h_{11},v_{1}\alpha,v_{1}h_{21}$ survive to $E_{4}(End(M);Q_{1})$. Furthermore,

$d_{2}(v_{1})=\alpha h_{11}^{2}$

$d_{3}(v_{1}^{2})=h_{11}^{3}$

Proof of Theorem 1:

Since we will need to distinguish between differentials in $E_{r}(End(M);Q_{1})$ and $E_{r}(M;Q_{1})$, we will denote them by $d_{r}$ and $d_{r}^{M}$ respectively.

In $E_{r}(M;Q_{1})$, $h_{11}^{3}$ must be a coboundary at some point and for degree reasons $d_{3}^{M}(v_{1}^{2})=h_{11}^{3}$. Indeed, if $d_{r}(x)=h_{11}^{3}$ for some $r\geq 3$ and $x\in E_{r}(M;Q_{1})$ then since $|h_{11}^{3}|=(3,6,0)$ and $d_{r}^{M}$ changes degrees by $(r,r-1,1-r)$ we conclude that $|x|=(3-r,7-r,r-1)$. Recall $|v_{1}|=(0,2,1),\,|\alpha|=(0,-1,0),\,|h_{n1}|=(1,2^{n+1}-2,0)$. Then $3-r\geq 0$, so $r=3$ and $|x|=(0,4,2)$. The only option now is $x=v_{1}^{2}$. Note if $v_{1}$ was to survive to $E_{3}(End(M);Q_{1})$ then $d_{3}(v_{1}^{2})=0$, which would force $d_{3}^{M}(v_{1}^{2})=0$. Hence $d_{2}(v_{1})\neq 0$ and so for degree reasons $d_{2}(v_{1})=\alpha h_{11}^{2}$. Given the action of $E_{2}(End(M);Q_{1})$ we must also have $d_{2}(v_{1})=\alpha h_{11}^{2}$. Either of those differentials could be also seen since $d_{2}(v_{1})=h_{10}h_{11}$ in $E_{2}(S;Q_{1})$ which follows from the same differential in the Cartan-Eilenberg spectral sequence computing $H^{*}(A(1))$.

Next we claim $d_{2}(h_{21})\neq 0$. Indeed, assume that $d_{2}(h_{21})=0$. Then $d_{2}(v_{1}^{2}h_{21})=0$ and since $v_{1}^{2}h_{21}$ survives in $E_{r}(M;Q_{1})$ it must be that $d_{3}(v_{1}^{2}h_{21})=0$ in $E_{3}(End(M);Q_{1})$. By multiplicativity we conclude $d_{3}(h_{21})=v_{1}^{-2}h_{11}^{3}h_{21}$. But now considering the action $E_{3}(End(M);Q_{1})\otimes E_{3}(M;Q_{1})\to E_{3}(M;Q_{1})$ we have

[TABLE]

which can’t happen since $h_{21}v_{1}$ survives in $E_{r}(M;Q_{1})$. Note we have to consider the action since $h_{21}v_{1}$ would not be present in $E_{3}(End(M);Q_{1})$. Hence our assumption was wrong and $d_{2}(h_{21})\neq 0$, which by degree reasons means $d_{2}(h_{21})=v_{1}^{-1}\alpha h_{11}^{2}h_{21}$.

Finally both $h_{11}$ and $v_{1}h_{21}$ survive $d_{3}^{M}$ in $E_{3}(M;Q_{1})$, so they must also survive $d_{3}$ in $E_{3}(End(M);Q_{1})$ i.e. $d_{3}(h_{11})=d_{3}(v_{1}h_{21})=0$. At the same time, for degree reasons $d_{r}(\alpha)=d_{r}(\alpha v_{1})=0$ for $r=2,3$ and neither elements can be a coboundary, which means both $\alpha$ and $\alpha v_{1}$ are present in $E_{4}(End(M);Q_{1})$.

$\Box$

Given the theorem above, in order to compute $d_{2}$ completely we just need to know the values on the remaining generators i.e. $d_{2}(h_{n1})$ for $n\geq 3$. Thus we make the following conjecture:

[TABLE]

.

Observe then $x_{n}=v_{1}h_{n+1,1}$ is a cycle, and that

[TABLE]

where the first factor has zero differential and the second factor has only $d_{2}v_{1}=\alpha h_{11}^{2}$. The homology is thus

[TABLE]

where $\alpha^{\prime}$ is the class of $v_{1}\alpha$. Again Theorem 1 tells us $d_{3}(x_{1})=d_{3}(\alpha)=d_{3}(\alpha^{\prime})=0$ and $d_{3}(v_{1}^{2})=h_{11}^{3}$ and so in order to compute $d_{3}$ completely we just need to know the values on the remaining generators i.e. $d_{3}(x_{n})$ for $n\geq 2$. Thus we further conjecture:

[TABLE]

We can prove this conjecture modulo the following assumption:

(Smaller) conjecture: $v_{1}^{m}x_{n}$ does not survive to $E_{4}(End(M);Q_{1})$ for $n,m\in\mathbb{Z}$, $n\geq 2$.

Theorem 2: The smaller conjecture above implies the main one.

Before proving the Theorem observe the converse statement that the main conjecture implies the smaller one also holds. In fact, the main conjecture even specifies what $d_{r}(v_{1}^{m}x_{n})$ is, which is what justifies the naming convention of the two conjectures. Thus, the Theorem can be reformulated by saying that the smaller and main conjectures above are equivalent.

Proof of Theorem 2:

For $n\geq 3$ $d_{2}(h_{n1})$ is a linear combination of $v_{1}^{-1}\alpha h_{11}^{2}h_{n1}$ and $v_{1}^{-1}\alpha h_{21}h_{n-1,1}^{2}$ for degree reasons, but the later is not in the image of $E_{2}(S;Q)$. Hence $d_{2}(h_{n1})=v_{1}^{-1}\alpha h_{11}^{2}h_{n1}$ or [math]. Assume that for some $n\geq 3$ $d_{2}(h_{n1})=0$. For degree reasons, $d_{3}(h_{n1})$ is a linear combination of $v_{1}^{-2}h_{11}^{3}h_{n1}$ and $v_{1}^{-2}h_{11}h_{21}h_{n-1,1}^{2}$, but $v_{1}^{-2}h_{11}h_{21}h_{n-1,1}^{2}$ doesn’t survive to $E_{3}(End(M);Q_{1})$ since

[TABLE]

By our smaller conjecture, $d_{3}(h_{n1})\neq 0$ and so $d_{3}(h_{n1})=v_{1}^{-2}h_{11}^{3}h_{n1}$. Then

[TABLE]

which again contradicts the (smaller) conjecture. We conclude $d_{2}(h_{n1})=v_{1}^{-1}\alpha h_{11}^{2}h_{n1}$ for all $n\geq 2$, which is also equivalent to $d_{2}(v_{1}h_{n1})=0$ for all $n\geq 2$. Hence the elements $x_{n}=v_{1}h_{n+1,1}$ survive, which justifies their presence in $E_{3}$. This completes the $d_{2}$ calculation in $E_{2}(End(M);Q_{1})$.

Next for $n\geq 2$ $d_{3}(x_{n})$ is a linear combination of $v_{1}^{-4}h_{11}x_{1}x_{n-1}^{2}$ and $v_{1}^{-2}h_{11}^{3}x_{n}$, which leaves us with 4 possibilities. $d_{3}(x_{n})=v_{1}^{-2}h_{11}^{3}x_{n}$ would imply $d_{3}(v_{1}^{2}x_{n})=0$ and so $d_{3}(x_{n})=0$ or $v_{1}^{-2}h_{11}^{3}x_{n}$ are both ruled out as possibilities due to the (smaller) conjecture. Then either $d_{3}(x_{n})=v_{1}^{-4}h_{11}x_{1}x_{n-1}^{2}$ or $d_{3}(x_{n})=v_{1}^{-4}h_{11}x_{1}x_{n-1}^{2}+v_{1}^{-2}h_{11}^{3}x_{n}$. However, the latter case would imply

[TABLE]

and so

[TABLE]

which is false as $v_{1}^{-4}h_{11}^{4}x_{1}x_{n-1}^{2}$ is present in $E_{3}(End(M);Q_{1})$. We conclude $d_{3}(x_{n})=v_{1}^{-4}h_{11}x_{1}x_{n-1}^{2}$ for $n\geq 2$ as desired.

$\Box$

It is worth mentioning that Palmieri’s original conjecture would imply that $d_{3}^{M}(v_{1}^{m}h_{n1})\neq 0$ for $n\geq 3$, which would guarantee the (smaller) conjecture. However, the smaller conjecture itself is enough to arrive at a different answer than what Palmieri suggested. This proves his original formulation is incorrect, but as we will see in the next section it is close to what we arrive at based on the (smaller) conjecture.

4.1 Completing the calculation of $d_{3}$ in $E_{3}(M;Q_{1})$

Now that we have learnt a fair bit about the structure of $E_{r}(End(M);Q_{1})$ we will see how the information about its differentials can translate to information about the differentials in $E_{r}(M;Q_{1})$. Recall for degree reasons $E_{2}(M;Q_{1})=E_{3}(M;Q_{1})$. Observe $E_{3}(M;Q_{1})$ is now generated by $\{1,v_{1}\}$ as a $E_{3}(End(M);Q_{1})$-module. Since $v_{1}$ survives to $E_{\infty}(M;Q_{1})$ we get $d_{3}^{M}(v_{1})=d_{3}^{M}(1)=0$ and so $d_{3}$ now completely determines $d_{3}^{M}$.

For example, to compute $d_{3}^{M}(h_{n1})$ for $n\geq 3$ note that $h_{n1}=v_{1}^{-2}x_{n-1}\cdot v_{1}$ and so we get

[TABLE]

We conclude that assuming the (smaller) conjecture holds, the differentials in $E_{3}(M;Q_{1})$ are

$d_{3}^{M}(v_{1}^{2})=h_{11}^{3}$

$d_{3}^{M}(h_{21})=v_{1}^{-2}h_{21}h_{11}^{3}$

$d_{3}^{M}(h_{n1})=v_{1}^{-2}h_{11}^{3}h_{n1}+v_{1}^{-2}h_{11}h_{21}h_{n-1,1}^{2}$ for $n\geq 3$

which is what we conjectured in Section 2.

5 Relation between Palmieri’s and Mahowald’s notations

In this section we will see how the conjectured differentials for $E_{3}(M;Q_{1})$ imply Mahowald’s conjecture assuming there are no higher degree differentials. We begin by stating Mahowald’s conjecture explicitly following the original description in [1]. Let $P=\mathbb{F}_{2}[x_{1},x_{2},\cdots]$ be a polynomial algebra, which is bigraded with $|x_{i}|=(2,2^{i+2}+1)$. Set a derivation $d$ on $P$ by $d(x_{i})=x_{1}x_{i-1}^{2}$ for $i>1$. Let $H(d)$ be the resulting homology and $B(d)$ the image of $d$. Then assuming $a$ and $b$ run through an $\mathbb{F}_{2}$-basis for $H(d)$ and $B(d)$ Mahowald conjectured that

[TABLE]

Here $bo$ and $bu$ are connective real and complex $K$-theory respectively and we have explicit computations:

[TABLE]

[TABLE]

In other words, the conjecture reads that $v_{1}^{-1}E_{2}(M;H)$ consists of $|H(d)|$ copies of $\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}(h_{11},v_{1})/(h_{11}^{3},v_{1}^{2})$ and $|B(d)|$ copies of $\mathbb{F}_{2}[v_{1}^{\pm 1}]$. To clarify, by $|H(d)|$ we mean the number of basis elements of any given degree in $H(d)$ and even though $H(d)$ is infinite, it is of finite type and so for every basis element $a\in H(d)$ the copy is suspended by the degree of $a$. The same holds for $B(d)$.

Recall $E_{3}=E_{3}(M;Q_{1})=\mathbb{F}_{2}[v_{1}^{\pm 1}]\otimes\mathbb{F}_{2}[h_{11},h_{21},h_{31},...]$ with proposed differentials $d_{3}(v_{1}^{2})=h_{11}^{3}$ and $d_{3}(h_{n1})=v_{1}^{-2}h_{11}^{3}h_{n1}+v_{1}^{-2}h_{11}h_{21}h_{n-1,1}^{2}$ for $n>2$. We will express $E_{4}$ in such a way that it takes the form Mahowald suggested. Rewrite $E_{3}=\mathbb{F}_{2}[v_{1}^{\pm 1},h_{11}]\otimes\mathbb{F}_{2}[x_{1},x_{2}...]$ where $x_{n}=v_{1}h_{n+1,1}$ and introduce a grading on $E_{3}$ so that |v_{1}^{i}|=\begin{cases}\begin{array}[]{cc}0&\text{if }i\equiv 0,1(4)\\ 2&\text{if }i\equiv 2,3(4)\end{array}\end{cases}, $|h_{11}|=1$ and $|x_{n}|=0$. Extend this grading to monomials in the obvious fashion. Then $E_{3}=\oplus_{n\geq 0}E_{3,n}.$ The reason we are interested in this grading is that now $d_{3}$ increases it by $1$. But then $E_{4}$ is just the homology of the graded chain complex i.e. $E_{4}=\oplus_{n\geq 0}\ker(d_{3}^{n})/\text{im}(d_{3}^{n-1})$.

[TABLE]

We claim that

$(1)$ $\ker(d_{3}^{0})/\text{im}(d_{3}^{-1})=\ker(d_{3}^{0})=Z(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})$

$(2)$ $\ker(d_{3}^{1})/\text{im}(d_{3}^{0})=H(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{h_{11}\}$

$(3)$ \begin{array}[]{c}\Big{(}\ker(d_{3}^{2})/\text{im}(d_{3}^{1})\Big{)}/H(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{h_{11}^{2}\}\cong\\ \cong B(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{v_{1}^{2}\}\end{array}

$(4)$ $\ker(d_{3}^{n})/\text{im}(d_{3}^{n-1})=0$ for $n\geq 3$

Given the proof of $(1)-(4)$ is not particularly insightful, we leave it for the end of this section. We are left with the task of identifying the expressions above with Mahowald’s formulation. The key here is to observe that given $(2)$ and $(3)$ we would need to identify $Z(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})$ in $(1)$ with $(H(d)\oplus B(d))\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})$. Then from $(1),(2),(3)$ we would get the $|H(d)|$ copies of $\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}(h_{11},v_{1})/(h_{11}^{3},v_{1}^{2})$. What is left over is $B(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})$ from $(1)$ and $B(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{v_{1}^{2}\}$ from $(3)$, which combine to produce $|B(d)|$ copies of $\mathbb{F}_{2}[v_{1}^{\pm 1}]$. Thus each of $(1),(2)$ and $(3)$ corresponds to a third of the “lightning flash” sequence, while the remainder of $(1)$ and $(3)$ each represent half of the $v_{1}$-line.

Below we can see exactly how the elements of $H(d)$ and $B(d)$ correspond to lightning flashes and $v_{1}$-lines in $E_{2}(M;H)$. The first few elements of $H(d)$ appearing are $1,x_{1},x_{1}^{2},x_{2}^{2}$ and $x_{1}^{2}x_{3}+x_{2}^{3}$ and we can see the lightning falshes for each one. Similarly, the first few elements of $B(d)$ appearing are $x_{1}^{3}$ through $x_{1}^{9}$ and $x_{1}x_{2}^{2}$ each corresponding to a copy of $\mathbb{F}_{2}[v_{1}^{\pm 1}]$. The colors used have no underlying meaning outside of grouping together the different elements in $E_{2}(M;H)$ and relating each group to its representing element of $H(d)$ or $B(d)$.

We are left to prove $(1)-(4)$. It is an immediate check to verify they follow from $(i)$ and $(ii)$ below, which is what we set out to show.

[TABLE]

[TABLE]

Note that that $E_{3}^{0}=P\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})$, $E_{3}^{1}=P\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{h_{11}\}$ and $d_{3}^{0}(y)=d(y)v_{1}^{-4}h_{11}$ for every $y\in P\subset E_{3}^{0}$. Hence $\text{ker}(d_{3}^{0})$ and $\text{im}(d_{3}^{0})$ take the desired form and the same argument holds for $\text{ker}(d_{3}^{1})$ and $\text{im}(d_{3}^{1})$. We proceed to calculate $\ker(d_{3}^{2})$ and the calculation of $\ker(d_{3}^{n})$ for $n>2$ is analogous. Every element of $E_{3}^{2}$ takes the form $\sum_{i=1}^{s}v_{1}^{m_{i}}y_{i}+\sum_{j=1}^{t}v_{1}^{l_{j}}z_{j}h_{11}^{2}$ where $m_{1}<m_{2}<\cdots<m_{s}$, $m_{i}\equiv 2,3(4)$, $l_{1}<l_{2}<\cdots l_{t},$ $l_{j}\equiv 0,1(4)$ and $y_{i},z_{j}\in P$. We also assume $y_{i},z_{j}\neq 0$. Then

[TABLE]

Setting this equal to [math] we observe two cases. First if $s=0$ then $d(z_{j})=0$ for all $j$ and we get the same component as in $\ker(d_{3}^{0})$, namely $Z(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{h_{11}^{2}\}\subset\ker(d_{3}^{2})$. If $s>0$ then we obtain $d(y_{i})=0$ for all $i$ and we are left with

[TABLE]

which given the degrees of $v_{1}$ can only happen if $s=t$, $m_{i}-2=l_{i}-4$ and $y_{i}=d(z_{i})$. Note $y_{i}=d(z_{i})$ already implies $d(y_{i})=0$. Furthermore, for every $y_{i}\in B(d)$ we have a unique $z_{i}\in P$ with $y_{i}=d(z_{i})$ modulo $Z(d)\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})\otimes\{h_{11}^{2}\}\subset\ker(d_{3}^{2})$. Hence

[TABLE]

as desired. In fact, $\ker(d_{3}^{2})\cong P\otimes\mathbb{F}_{2}[v_{1}^{\pm 4}]\otimes\mathbb{F}_{2}[v_{1}]/(v_{1}^{2})$, but stated this way it does not relate well with Mahowald’s conjecture.

Next we show $\text{im}(d_{3}^{2})=\ker(d_{3}^{3})$ and the result for $\text{im}(d_{3}^{n})$ follows analogically. As we saw above elements of $\ker(d_{3}^{3})$ are sums of elements of the form $v_{1}^{m}yh_{11}+v_{1}^{m-2}zh_{11}^{3}$ for $m\equiv 2,3(4)$ and $y,z\in P$ such that $d(z)=y$. But then $d_{3}^{2}(v_{1}^{m}z)=v_{1}^{m}yh_{11}+v_{1}^{m-2}zh_{11}^{3}$ and so $\ker(d_{3}^{3})\subset\text{im}(d_{3}^{2})$ and since the reverse inclusion holds as well the two must coincide. This completes the proof of $(i)$ and $(ii)$ and thus we have successfully identified Mahowald’s and Palmieri’s formulations of the problem.