Fixed-point index, the Incompatibility Theorem, and torus parametrization

TL;DR

This paper generalizes the fixed-point index concept for Jordan curves, introduces a new torus parametrization tool, and provides topological proofs of key theorems related to circle packings and homeomorphisms.

Contribution

It extends the Circle Index Lemma, introduces torus parametrization for fixed-point index analysis, and offers a topological proof of the Incompatibility Theorem.

Findings

Generalized the Circle Index Lemma for non-disconnecting Jordan curves

Introduced torus parametrization as a combinatorial tool for fixed-point index problems

Provided the first purely topological proof of a key lemma used in circle packing theorems

Abstract

The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round circles is always non-negative. We begin by proving a generalization of this lemma, to accommodate Jordan curves bounding domains which do not disconnect each other. We then apply this generalization to give a new proof of Schramm's Incompatibility Theorem, which was used by Schramm to give the first proof of the rigidity of circle packings filling the complex and hyperbolic planes. As an example application, we include outlines of proofs of these circle packing theorems. We then introduce a new tool, the so-called torus parametrization, for working with fixed-point index, which allows some problems concerning this quantity to be approached combinatorially. We apply torus parametrization to give the first purely topological proof of the following lemma: given two positively oriented Jordan curves, one may essentially prescribe the images of three points of one of the curves in the other, and obtain an orientation-preserving homeomorphism between the curves, having non-negative fixed-point index, which respects this prescription. This lemma is essential to our proof of the Incompatibility Theorem.