On Bruner's Open Questions: Secondary Ext of the Fibe of $S{q}^n$ 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.

26 pages. This version corrects and restructures the results presented in v1, and the title of the manuscript has been revised accordingly. The author would be grateful for any comments