The Goldman bracket characterizes homeomorphisms between non-compact surfaces

TL;DR

This paper establishes that for certain non-compact surfaces, a homotopy equivalence is homotopic to a homeomorphism precisely when it preserves the Goldman bracket, excluding the plane and punctured plane cases.

Contribution

It characterizes when a homotopy equivalence between non-compact surfaces can be homotoped to a homeomorphism using the Goldman bracket.

Findings

Homotopy equivalences preserving the Goldman bracket are homotopic to homeomorphisms.

The result applies to all non-compact orientable surfaces except the plane and punctured plane.

Provides a new criterion for surface homeomorphism classification.

Abstract

We show that a homotopy equivalence between two non-compact orientable surfaces is homotopic to a homeomorphism if and only if it preserves the Goldman bracket, provided our surfaces are neither the plane nor the punctured plane.