On the $h$-cobordism category. I

TL;DR

This paper investigates the topological category of h-cobordisms between manifolds with boundary and compares its homotopy type to the standard h-cobordism space of a compact smooth manifold, providing insights into their relationship.

Contribution

It introduces a comparison between the homotopy type of the h-cobordism category and the classical h-cobordism space, advancing understanding of their topological structures.

Findings

Homotopy type of the h-cobordism category is characterized.

Comparison reveals new relationships between categories and spaces.

Results contribute to the understanding of manifold topology.

Abstract

We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.