Mosc. Math. J., 2025 Volume 25, Number 2,Pages 163–196(Mi mmj906)
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 $\mathcal A_2$
Abstract:
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.
Key words and phrases:group actions on combinatorial structures, invariant ring, Steenrod algebra, hit problem, algebraic transfer.
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