Illinois Journal of Mathematics

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

Andrey M. Mishchenko

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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.

Article information

Illinois J. Math., Volume 60, Number 2 (2016), 413-445.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]
Secondary: 52C26: Circle packings and discrete conformal geometry


Mishchenko, Andrey M. Fixed-point index, the Incompatibility Theorem, and torus parametrization. Illinois J. Math. 60 (2016), no. 2, 413--445. doi:10.1215/ijm/1499760015.

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  • E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. 83 (1970), no. 125, 256–260 (Russian).
  • B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012.
  • Z.-X. He and O. Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2) 137 (1993), no. 2, 369–406.
  • P. Koebe, Kontaktprobleme der Konformen Abbildung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 88 (1936), 141–164. (German).
  • S. Merenkov, Planar relative Schottky sets and quasisymmetric maps, Proc. Lond. Math. Soc. (3) 104 (2012), no. 3, 455–485.
  • A. M. Mishchenko, Rigidity of thin disk configurations, Ph.D. dissertation, University of Michigan, Ann Arbor, 2012; available at
  • A. M. Mishchenko, Rigidity of thin disk configurations, via fixed-point index, 2013, preprint; available at \arxivurlarXiv:1302.2380 [math.MG].
  • S. Rohde, Oded Schramm: From circle packing to SLE, Ann. Probab. 39 (2011), 1621–1667.
  • H. Sachs, Coin graphs, polyhedra, and conformal mapping, Discrete Math. 134 (1994), no. 1–3, 133–138.
  • O. Schramm, Rigidity of infinite (circle) packings, J. Amer. Math. Soc. 4 (1991), 127–149.
  • K. Stephenson, Introduction to circle packing: The theory of discrete analytic functions, Cambridge University Press, Cambridge, 2005.
  • K. Strebel, Über das Kreisnormierungsproblem der konformen Abbildung, Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 1951 (1951), no. 101, \bnumber22 (German).
  • W. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University, unpublished lecture notes, version 1.1, 1980; available at