On CW-complexes over groups with periodic cohomology

Abstract

If $G$ has $4$-periodic cohomology, then 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 ${\pi }_1(X)=G$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $\mathbb{Z}G$ modules where $G$ has periodic cohomology.

27 pages. Added new Corollary relating to a conjecture of J. M. Cohen in final section, small changes to layout and references, fixed minor errors. Final version, to appear in Transactions of the American Mathematical Society

References in corpus (2)

• Realizing algebraic 2-complexes by cell complexes
• An exotic presentation of Q_28

Cited by in corpus (4)

• An exotic presentation of Q_28
• Projective modules and the homotopy classification of $(G,n)$-complexes
• Homotopy classification of 4-manifolds whose fundamental group is dihedral
• Four-manifolds, two-complexes and the quadratic bias invariant