Abstract

In previous work, we related homotopy types of finite (G, n)-complexes when G has periodic cohomology to projective ZG-modules representing the Swan finiteness obstruction. We use this to determine when X ∨ Sn Y ∨ Sn implies X 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 ZGmodules 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.