A composition formula for manifold structures

Abstract

The structure set $\ST^{TOP}(M)$ of an $n$-dimensional topological manifold $M$ for $n \geqslant 5$ has a homotopy invariant functorial abelian group structure, by the algebraic version of the Browder-Novikov-Sullivan-Wall surgery theory. An element $(N,f) \in \ST^{TOP}(M)$ is an equivalence class of $n$-dimensional manifolds $N$ with a homotopy equivalence $f:N \to M$. The composition formula is that $(P,fg)=(N,f)+f_*(P,g) \in \ST^{TOP}(M)$ for homotopy equivalences $g:P \to N$, $f:N \to M$. The formula is required for a paper of Kreck and L\"uck.