2025/07/29 by Hofmann, Tommy, Nicholson, John
#20C05 #55P15 #57Q12 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.2507.21975
We study Swan modules, which are a special class of projective modules over integral group rings, and their consequences for the homotopy classification of CW-complexes. We show that there exists a non-free stably free Swan module, thus resolving Problem A4 in the 1979 Problem List of C. T. C. Wall. As an application we show that, in all dimensions n ≡ 3 mod 4, there exist finite n-complexes which are homotopy equivalent after stabilising with multiple copies of Sn, but not after a single stabilisation. This answers a question of M. N. Dyer. We also resolve a question of S. Plotnick concerning Swan modules associated to group automorphisms and, as an application, obtain a short and direct proof that there exists a group with k-periodic cohomology which does not have free period k. In contrast to the original proof our R. J. Milgram, our proof circumvents the need to compute the Swan finiteness obstruction.