Equivariant Moore spaces and the Dade group

TL;DR

This paper studies equivariant Moore spaces for finite p-groups, showing their homology modules are endo-permutation modules generated by relative syzygies, and relates the Grothendieck group of these spaces to the Dade group.

Contribution

It introduces the concept of $ ext{ extunderscore}n$-Moore $G$-spaces and establishes their homology modules as endo-permutation modules generated by relative syzygies, linking topology and representation theory.

Findings

Homology modules of finite Moore $G$-spaces are endo-permutation modules.

The Grothendieck group of Moore $G$-spaces relates to the Dade group.

Fixed point sets are $ ext{ extunderscore}n(H)$-Moore spaces for subgroups $H$.

Abstract

Let $G$ be a finite $p$-group and $k$ be a field of characteristic $p$. A topological space $X$ is called an $n$-Moore space if its reduced homology is nonzero only in dimension $n$. We call a $G$-CW-complex $X$ an $\underline{n}$-Moore $G$-space over $k$ if for every subgroup $H$ of $G$, the fixed point set $X^H$ is an $\underline{n}(H)$-Moore space with coefficients in $k$, where $\underline{n}(H)$ is a function of $H$. We show that if $X$ is a finite $\underline{n}$-Moore $G$-space, then the reduced homology module of $X$ is an endo-permutation $kG$-module generated by relative syzygies. A $kG$-module $M$ is an endo-permutation module if ${\rm End}_k (M) =M \otimes _{k} M^*$ is a permutation $kG$-module. We consider the Grothendieck group of finite Moore $G$-spaces $\mathcal{M}(G)$, with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies.