Moscow Mathematical Journal
Volume 25, Issue 2, April–June 2025 pp. 163–196.
On admissible $\mathcal A_2$-generators for the cohomology ring $H^*((G_1(\mathbb R^{\infty}))^{\times t}; \mathbb Z_2)$ and the (mod-2) cohomology of the Steenrod algebra $\mathcal A_2$
It is well known that the “hit problem” is an important problem in
algebraic topology, which involves determining a minimal generating
set for a specific module related to the Steenrod algebra. While
notable progress has been made for small cases, the general problem
remains unsolved, particularly for larger numbers of variables. A
related application in this study is to describe the Singer
cohomological transfer, which provides insights into the structure of
the (mod-2) cohomology groups of the Steenrod algebra. Nonetheless,
these cohomology groups remain poorly understood in higher homological
degrees. In this work, we strengthen results for the hit problem with
five or more variables in certain generic degrees and analyze the
behavior of the Singer transfer in the relevant
bidegrees. Additionally, we provide a set of efficient,
computer-assisted algorithms — implementable in SageMath and
Maple — that effectively address various aspects of the hit
problem and the Singer transfer. 2020 Math. Subj. Class. 05E18, 13A50, 55S10, 55R12.
Authors:
Đặng Võ Phúc (1)
Author institution:(1) Department of Information Technology, FPT University, Quy Nhon A.I Campus, An Phu Thinh New Urban Area, Quy Nhon City, Binh Dinh, Vietnam
Summary:
Keywords: Group actions on combinatorial structures, invariant ring, Steenrod algebra, hit problem, algebraic transfer.
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