Equivariant Moore spaces and the Dade group

By Ergün Yalçın. 2016

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/729ae2e3d834b9bb51f774f8e8cb412b10c46b6f

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