Hyperbolic Geometry and Homotopic Homeomorphisms of Surfaces

TL;DR

This paper introduces a new hyperbolic geometric approach to prove that homotopic homeomorphisms of surfaces are isotopic, extending classical results to a broader class of surfaces with minimal exceptions.

Contribution

It provides a novel method connecting hyperbolic geometry with curve isotopy theory to establish the isotopy of homotopic homeomorphisms on most surfaces.

Findings

New proof method using hyperbolic geometry

Extension of Baer's theorem to non-orientable and non-compact surfaces

Identification of 13 exceptional surfaces requiring special proofs

Abstract

The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A. Epstein to arbitrary surfaces, compact or not, with or without boundary and orientable or not. We give a new method of deducing the theorem about homotopic homeomorphisms from the results about homotopic curves via the hyperbolic geometry of surfaces. This works on all but 13 surfaces where ad hoc proofs are needed.