Adams filtrations and secondary coherence for fibers of Steenrod operations

Abstract

Motivated by the Bruner--Rognes two-extension theorem and open questions regarding the secondary cohomology modeling of fibers of Steenrod operations, we develop a general framework to compute their Adams filtrations and delineate the underlying algebraic obstructions. Let $f:Y\to \prod {\Sigma }^{d_{\lambda }}H{\mathbb{F}}_2$ be a locally finite family of positive-degree stable operations, where $Y=H{\mathbb{F}}_2$ or $H\mathbb{Z}$. We determine the Adams filtration of its fiber from the minimal generators of $\mathrm{im}(f^{\ast })$ as a left Steenrod module. An explicit target automorphism separates a minimal subfamily, and combining this with the two-extension theorem and an acyclic cancellation gives $E_3=E_{\infty }$. A terminating Adem recursion realizes this splitting for the full positive even-square family. Separately, we identify two fundamental algebraic obstructions to directly constructing the secondary cohomology of these fibers: an augmentation discrepancy when the primary row has a nonzero kernel, and a coherence obstruction class in the third cohomology of a Hom complex. These results demonstrate why the secondary Ext groups of these fibers must be derived topologically via the Baues--Jibladze comparison theorem, rather than through a direct algebraic resolution.

arXiv comment21 pages. This new version supersedes v2 and withdraws its unsupported claims to resolve Bruner's Questions 6.1 and 6.2. Instead, it rigorously delineates the structural obstructions that make a direct algebraic secondary modeling of these fibers impossible