On Bruner's Open Questions: Secondary Ext of the Fibe of $Sq^n$ via Explicit Secondary Adem Tracks

Abstract

Robert Bruner \cite[Questions 6.1 and 6.2]{Bruner2026} asked whether the secondary cohomology of the fibers $F_n$, $F_{n\mathbb{Z}}$, and $F$ can be computed to determine the $E_3$-terms of their Adams spectral sequences, and whether the Bruner-Rognes two-extension formula for the ordinary Adams $d_2$ is intrinsic to secondary cohomology. In this work, we give an unconditional affirmative answer to both questions. Working in the Baues-Nassau secondary Steenrod algebra, we construct explicit secondary mapping-fiber resolutions for these fibers using a tracked Adem reduction algorithm and the Baues-Jibladze recursive completion. We determine the secondary Ext groups, independently recovering Bruner's $E_3$-terms: \[ \operatorname{Ext}_{\mathcal{B}}^{*,*}(H_{\mathcal{B}}^* F_n, \mathbb{F}_2) \cong \mathbb{F}_2 \oplus Σ^{1,n}\mathbb{F}_2, \] \[ \operatorname{Ext}_{\mathcal{B}}^{*,*}(H_{\mathcal{B}}^* F_{n\mathbb{Z}}, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus Σ^{1,n}\mathbb{F}_2, \] \[ \operatorname{Ext}_{\mathcal{B}}^{*,*}(H_{\mathcal{B}}^* F, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus \bigoplus_{j>0} Σ^{1,2^j}\mathbb{F}_2 \oplus \bigoplus_{\substack{i>0 \\ i \text{ not a power of } 2}} Σ^{0,2i-1}\mathbb{F}_2. \] This direct calculation answers Question 6.1. Finally, we prove that the primary shadow of the first secondary differential in our construction is identically the Bruner-Rognes Yoneda composite associated with the corresponding two-extension, thereby answering Question 6.2.