Functoriality of Isovariant Homotopy Classification

TL;DR

This paper extends the functoriality of homotopy classification from topological manifolds to G-equivariant manifolds with isovariant structures, linking it to the Farrell-Jones Conjecture for L-theory.

Contribution

It introduces the isovariant structure set for G-equivariant manifolds and shows its functoriality and relation to the assembly map and Farrell-Jones Conjecture.

Findings

Isovariant structure set is functorial under equivariant maps.

The structure set is the fiber of the assembly map for L-homology.

Connects isovariant classification to the Farrell-Jones Conjecture.

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.