Cancellation for (G,n)-complexes and the Swan Finiteness Obstruction

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 that lead to a new approach to finding a counterexample to Wall’s D2 problem.