Functoriality of Isovariant Homotopy Classification

2009/09/28 by Cappell, Sylvain, Weinberger, Shmuel, Yan, Min
#55N91 #57N80 #57R67 #FOS: Mathematics #Geometric Topology (math.GT) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.0909.5005

Abstract

It is a deep fact that the homotopy classification of topological manifolds is convariantly functorial. In other words, a map from a topological manifold M to another N naturally induces a map from the structure set S(M) to S(N). We extend the fact to the isovariant structure set S_G(M, rel M_s) of G-equivariant topological manifolds isovariantly homotopy equivalent to M and restricts to homormorphism on the singular part M_s, consisting of those points fixed by some non-trivial elements of G. We further explain that the structure set S_G(M, rel M_s) is the fibre of the assembly map for the generalized homology theory with the L-spectrum as the coefficient. This relates our result to the Farrell-Jones Conjecture for L-theory.

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