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Structure of the space of GL4 (Z2)-coinvariants Z2 ⊗ GL4 (Z2) P H∗ (Z42,Z2) in some generic degrees and its application

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posted on 2023-08-23, 02:32 authored by Đặng Võ PhúcĐặng Võ Phúc

Fix $k = \mathbb Z_2$ to be a field of characteristic 2, let $A$ denote the Steenrod algebra over $k.$ A problem of immense difficulty in algebraic topology is the determination of a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_1|= |x_2| = \cdots = |x_q| =  1.$ By way of equivalence, one may choose to write an explicit basis for the cohit space $\pmb{Q}^{q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This subject, which has now a long history, is the content of the classical "hit problem" proposed in [Abstracts Papers Presented Am. Math. Soc.  \textbf{833} (1987), 55-89]. Furthermore, it is closely related to the $q$-th transfer homomorphism $Tr_q^{A}$ constructed by William Singer in [Math. Z. \textbf{202} (1989), 493-523]. This map $Tr_q^{A}$ passes from the space of $G(q)$-coinvariant $k\otimes _{G(q)} P_A((P_q)_n^{*})$ of $\pmb{Q}^{q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k),$ wherein $G(q)$ stands for the general linear group of degree $q$ over the field $k,$ and $P_A((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ Particularly, the assertion that $Tr_q^{A}$ is always an injective map has been conjectured by Singer himself, but as of now, this remains an open problem for all $q\geq 4.$ Accordingly, the aim of the present study is to deal with the Singer conjecture for rank 4 in certain internal degrees. Specifically, by the usage of the techniques of the hit problem in four variables, we explicitly determine the structure of the coinvariant $k\otimes _{G(4)} P_A((P_4)_{n}^{*})$ in some generic degrees $n.$ Then, applying these results and a representation of $Tr_4^{A}$ via the lambda algebra, we state that Singer's conjecture is true for rank $q = 4$ in respective degrees $n.$ This has significantly contributed towards the ultimate proof of Singer's conjecture within the rank 4 case.


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University of Khanh Hoa

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  • Viet Nam

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