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Summer 2016 Fixed-point index, the Incompatibility Theorem, and torus parametrization
Andrey M. Mishchenko
Illinois J. Math. 60(2): 413-445 (Summer 2016). DOI: 10.1215/ijm/1499760015


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.


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Andrey M. Mishchenko. "Fixed-point index, the Incompatibility Theorem, and torus parametrization." Illinois J. Math. 60 (2) 413 - 445, Summer 2016.


Received: 3 June 2013; Revised: 24 February 2017; Published: Summer 2016
First available in Project Euclid: 11 July 2017

zbMATH: 1377.54049
MathSciNet: MR3680541
Digital Object Identifier: 10.1215/ijm/1499760015

Primary: 54H25
Secondary: 52C26

Rights: Copyright © 2016 University of Illinois at Urbana-Champaign


Vol.60 • No. 2 • Summer 2016
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