  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.
