Projective modules and the homotopy classification of CW-complexes

Abstract: A basic question in the homotopy classification of CW-complexes is to ask for which finitely presented groups $G$ does $X \vee S^2 \simeq Y \vee S^2$ imply $X \simeq Y$, where $X$ and $Y$ are finite 2-complexes with fundamental group $G$. Despite early interest by Cockroft-Swan and Dyer-Sieradski, it wasn’t until 1976 that examples of non-cancellation were found by Dunwoody and Metzler. This led Browning to complete the classification in the finite abelian case. In recent years, applications to Wall’s D2 problem and the classification of manifolds have sparked renewed interest in this problem. In this talk, we will show how the case where $G$ has periodic cohomology can largely be reduced to a question about projective $\mathbb{Z} G$ modules. We then resolve this by generalising results of Swan from the 1980s.

algebraic topologydifferential geometrygeneral topologygroup theorygeometric topology

Audience: researchers in the topic

Ohio State Topology and Geometric Group Theory Seminar

Series comments: https://sites.google.com/view/topoandggt