On the functoriality of the space of equivariant smooth $h$-cobordisms

Abstract

We construct an $(\infty ,1)$-functor that takes each smooth $G$-manifold with corners $M$ to the space of equivariant smooth $h$-cobordisms ${\mathcal{H}}_{\mathrm{Diff}}(M)$. We also give a stable analogue ${\mathcal{H}}_{\mathrm{Diff}}^{\mathcal{U}}(M)$ where the manifolds are stabilized with respect to representation discs. The functor structure is subtle to construct, and relies on several new ideas. In the non-equivariant case $G=e$, our $(\infty ,1)$-functor agrees with previous constructions of the smooth $h$-cobordism space as a functor to the homotopy category.

91 pages. Thorough revision and improved exposition. Notably, the smooth extension and tubular neighborhood results for equivariant manifolds with corners are much more detailed now. We thank a very careful referee for many useful suggestions