Year: 2026
Type: preprint
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 $f:Y\to\prodΣ^{d_λ}H\mathbb F_2$ 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 $\operatorname{im}(f^*)$ 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.
Source: arXiv (Cornell University)
Author: Phuc, Dang Vo
Institution: FPT University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161803076