The Alexander trick for homology spheres

TL;DR

This paper proves that the group of boundary-fixing homeomorphisms of certain high-dimensional contractible manifolds is contractible, using a uniqueness result for one-sided h-cobordisms, advancing understanding in high-dimensional topology.

Contribution

It establishes the contractibility of the homeomorphism group for compact contractible manifolds in dimensions six and higher, based on a new uniqueness theorem for one-sided h-cobordisms.

Findings

Homeomorphism group is contractible for d ≥ 6

Strong uniqueness for one-sided h-cobordisms

Advances high-dimensional topological understanding

Abstract

We show that the group of homeomorphisms of a compact contractible $d$-manifold which fix the boundary is contractible, as long as $d \geq 6$. We deduce this from a strong uniqueness statement for one-sided $h$-cobordisms.