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

We construct an $(\mathrm{\infty },1)$-functor that takes each smooth $G$-manifold with corners $M$ to the space of equivariant smooth $h$-cobordisms ${\mathcal{H}}_{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}(M)$. We also give a stable analogue ${\mathcal{H}}_{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathcal{U}}(M)$ 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 $(\mathrm{\infty },1)$-functor agrees with previous constructions of the smooth $h$-cobordism space as a functor to the homotopy category.