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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
  • Đặng Võ Phúc
Đặng Võ Phúc
University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa

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Abstract

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.