Strong Topological Rigidity of Non-Compact Orientable Surfaces

TL;DR

This paper proves that all non-compact orientable surfaces, except the plane and punctured plane, are topologically rigid, meaning any proper homotopy equivalence is homotopic to a homeomorphism, highlighting their strong topological rigidity.

Contribution

It establishes a broad topological rigidity result for non-compact orientable surfaces, extending previous understanding of surface homotopy equivalences.

Findings

Proper homotopy equivalences are homotopic to homeomorphisms for most non-compact orientable surfaces.

The plane and punctured plane are exceptions to this rigidity.

All other non-compact orientable surfaces exhibit strong topological rigidity.

Abstract

We show that every orientable infinite-type surface is properly rigid as a consequence of a more general result. Namely, we prove that if a homotopy equivalence between any two non-compact orientable surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided surfaces are neither the plane nor the punctured plane. Thus all non-compact orientable surfaces, except the plane and the punctured plane, are topologically rigid in a strong sense.