On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring

Abstract

We write $\mathbb{P}$ for the polynomial algebra in one variable over the finite field ${\mathbb{Z}}_2$ and ${\mathbb{P}}^{\otimes t}={\mathbb{Z}}_2[x_1,\dots ,x_t]$ for its $t$-fold tensor product with itself. We grade ${\mathbb{P}}^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $({\mathbb{P}}^{\otimes t}{)}^{G_t}$ as a module over Steenrod ring, ${\mathcal{A}}_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t,{\mathbb{Z}}_2).$ An equivalent problem is to find a monomial basis of the space of "unhit" elements, ${\mathbb{Z}}_2{\otimes }_{{\mathcal{A}}_2}({\mathbb{P}}^{\otimes t}{)}^{G_t}$ in each $t$ and degree $n\ge 0.$ The structure of this tensor product is proved surprisingly difficult and has been not yet known for $t\ge 5,$ even for the trivial subgroup $G_t=\{e\}.$ In the present paper, we consider the subgroup $G_t=\{e\}$ for $t\in \{5,6\},$ and obtain some new results on ${\mathcal{A}}_2$-generators of $({\mathbb{P}}^{\otimes t}{)}^{G_t}$ in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [23, 25].

40 pages

References in corpus (5)

• On the generators of the polynomial algebra as a module over the Steenrod algebra
• Sub-Hopf algebras of the Steenrod algebra and the Singer transfer
• Lickorish type construction of manifolds over simple polytopes
• On the lambda algebra and Singer's cohomological transfer
• A note on the modular representation on the $\mathbb{Z}/2$-homology groups of the fourth power of real projective space and its application

Cited by in corpus (6)

• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four
• On the lambda algebra and Singer's cohomological transfer
• On Singer's conjecture for the fourth algebraic transfer in certain generic degrees
• The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees
• A note on the modular representation on the $\mathbb{Z}/2$-homology groups of the fourth power of real projective space and its application
• Structure of the space of $G{L}_4({\mathbb{Z}}_2)$-coinvariants ${\mathbb{Z}}_2{\otimes }_{G{L}_4({\mathbb{Z}}_2)}P{H}_{\ast }({\mathbb{Z}}_2^4,{\mathbb{Z}}_2)$ in some generic degrees and its application