Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction

Abstract

In previous work, we related homotopy types of finite $(G,n)$-complexes when $G$ has periodic cohomology to projective $\mathbb{Z}G$-modules representing the Swan finiteness obstruction. We use this to determine when $X\vee S^n\simeq Y\vee S^n$ implies $X\simeq Y$ for finite $(G,n)$-complexes $X$ and $Y$, and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective $\mathbb{Z}G$-modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise which lead to a new approach to finding a counterexample to Wall's D2 problem.

30 pages. Minor corrections made throughout. Further discussion on potential applications to Wall's D2 problem and the classification of 4-manifolds added. Final version, to appear in International Mathematical Research Notices