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

2020/05/04 by John William Nicholson, Nicholson, John · 2 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2005.01664

Abstract

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 \vee Sⁿ ≃ Y \vee 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 which lead to a new approach to finding a counterexample to Wall's D2 problem.

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• Exotic presentations of quaternion groups and Wall's D2 problem
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