Functoriality of Isovariant Homotopy Classification

It is a deep fact that the homotopy classification of topological manifolds is convariantly functorial. In other words, a map M → N of topological manifolds naturally induces a map S(M) → S(N) of their structure sets. We extend the fact to the isovariant structure set SG(M, rel Ms) of G-equivariant topological manifolds isovariantly homotopy equivalent to M and restricts to homormorphism on the singular part Ms, consisting of those points fixed by some non-trivial elements of G. We further explain that the structure set SG(M, rel Ms) 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.