On CW-complexes over groups with periodic cohomology

Nicholson, John

doi:10.1090/tran/8411

On CW-complexes over groups with periodic cohomology
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Trans. Amer. Math. Soc. 374 (2021), 6531-6557 Request permission

Abstract:

If $G$ has $4$-periodic cohomology, then finite D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Wall’s D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincaré $3$-complex $X$ with $G=\pi _1(X)$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $\mathbb {Z} G$ modules where $G$ has periodic cohomology.

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