Secondary Ext of the Fiber of $Sq^n$ and the Secondary Adams Spectral Sequence

TL;DR

This paper computes secondary Ext groups related to the fiber of Steenrod squares at the prime 2, advancing understanding of secondary cohomology and spectral sequences in algebraic topology.

Contribution

It provides explicit calculations of secondary Ext groups for specific fiber objects, connecting secondary algebraic structures with Adams spectral sequence differentials.

Findings

Explicit secondary Ext groups are determined for three fiber objects.

The calculations follow from Bruner's E_3 calculation and comparison theorems.

Secondary mapping fibers and Yoneda descriptions relate to Adams d_2 differentials.

Abstract

At the prime $2$, let $\mathcal{B}$ denote the secondary Steenrod algebra in the sense of Baues, Baues--Jibladze, Nassau, and Baues--Frankland. We determine the secondary Ext groups of the secondary cohomology objects of the three fibers considered by Bruner in his work on the fiber of $\mathrm{Sq}^n$. For $$F_n = \mathrm{fib}(H \xrightarrow{\mathrm{Sq}^n} \Sigma^n H),\,\, \text{and}\,\, F_{n\mathbb{Z}} = \mathrm{fib}(H\mathbb{Z} \xrightarrow{\mathrm{Sq}^n} \Sigma^n H),$$ where $n>0$ in the first case and $n \ge 2$ in the second, and for $$F = \mathrm{fib}(H\mathbb{Z} \to \prod_{i>0} \Sigma^{2i} H),$$ where the $i$-th component is represented after mod-$2$ reduction by $\mathrm{Sq}^{2i}$, the groups are $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_n, \mathbb{F}_2) \cong \mathbb{F}_2 \oplus \Sigma^{1,n}\mathbb{F}_2,$$ $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_{n\mathbb{Z}}, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus \Sigma^{1,n}\mathbb{F}_2,$$ and $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus \bigoplus_{j>0} \Sigma^{1,2^j}\mathbb{F}_2 \oplus \bigoplus_{i>0, i \neq 2^k} \Sigma^{0,2i-1}\mathbb{F}_2$$ (where the last direct sum runs over $i$ not a power of $2$). Here $h_0$ has bidegree $(1,1)$. The group-level calculation follows from Bruner's $E_3$-calculation and the Baues--Jibladze comparison theorem, and the proof gives an explicit primary calculation of the kernels and cokernels of the maps $D_n$, $D_{n\mathbb{Z}}$, and $D_F$. Separately, under an explicit track-fiber realization hypothesis, the same groups are computed from displayed secondary mapping fibers. Under the same hypothesis, the Bruner--Rognes Yoneda description of the ordinary Adams $d_2$ is the primary shadow of the first differential in secondary homological algebra.