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 Z G-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 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.