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

Abstract

In previous work, we related homotopy types of pG, nq-complexes when G has periodic cohomology to projective ZG modules representing the Swan finiteness obstruction. We use this to determine when X _ S n » Y _ S n implies X » Y for pG, nq-complexes X and Y , and give lower bounds for the number of minimal homotopy types of pG, nq-complexes when this fails. The proof involves constructing projective ZG 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.

A cancellation theorem for modules over integral group rings

“…We prove Theorem B, which involves establishing an analogue of Theorem E for groups which have a C 2 2 quotient (Section 8.3). Finally, we reflect on the applications of projective cancellation to topology given in [Nic21b,Nic20] in the case where G has periodic cohomology (Section 8.4). Our results lead to new and simpler proofs of the results obtained in those articles.…”