Algebraic & Geometric Topology

Relative fixed point theory

Kate Ponto

Full-text: Open access


The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative Lefschetz fixed point theorem and its converse.

Article information

Algebr. Geom. Topol., Volume 11, Number 2 (2011), 839-886.

Received: 1 November 2009
Revised: 2 December 2010
Accepted: 12 December 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25]
Secondary: 18D05: Double categories, 2-categories, bicategories and generalizations 55P25: Spanier-Whitehead duality

Reidemeister trace Nielsen theory fixed point Lefschetz number fixed point index trace bicategory


Ponto, Kate. Relative fixed point theory. Algebr. Geom. Topol. 11 (2011), no. 2, 839--886. doi:10.2140/agt.2011.11.839.

Export citation


  • C Bowszyc, Fixed point theorems for the pairs of spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968) 845–850
  • R F Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London (1971)
  • M Crabb, I James, Fibrewise homotopy theory, Springer Monogr. in Math., Springer, London (1998)
  • T tom Dieck, Transformation groups, de Gruyter Studies in Math. 8, de Gruyter, Berlin (1987)
  • A Dold, The fixed point transfer of fibre-preserving maps, Math. Z. 148 (1976) 215–244
  • A Dold, Lectures on algebraic topology, Classics in Math., Springer, Berlin (1995) Reprint of the 1972 edition
  • A Dold, D Puppe, Duality, trace, and transfer, from: “Proceedings of the International Conference on Geometric Topology (Warsaw, 1978)”, (K Borsuk, A Kirkor, editors), PWN, Warsaw (1980) 81–102
  • E Fadell, S Husseini, Fixed point theory for non-simply-connected manifolds, Topology 20 (1981) 53–92
  • S Husseini, Generalized Lefschetz numbers, Trans. Amer. Math. Soc. 272 (1982) 247–274
  • I M James, Fibrewise complexes, from: “Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guí xols, 1994)”, (C Broto, C Casacuberta, G Mislin, editors), Progr. Math. 136, Birkhäuser, Basel (1996) 193–199
  • J Jezierski, A modification of the relative Nielsen number of H Schirmer, Topology Appl. 62 (1995) 45–63
  • B J Jiang, Lectures on Nielsen fixed point theory, Contemporary Math. 14, Amer. Math. Soc. (1983)
  • B J Jiang, X Zhao, H Zheng, On fixed points of stratified maps, J. Fixed Point Theory Appl. 2 (2007) 225–240
  • J R Klein, B Williams, Homotopical intersection theory. I, Geom. Topol. 11 (2007) 939–977
  • J R Klein, B Williams, Homotopical intersection theory. II. Equivariance, Math. Z. 264 (2010) 849–880
  • L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer, Berlin (1986) With contributions by J E McClure
  • W Lück, Transformation groups and algebraic $K$–theory, Lecture Notes in Math. 1408, Springer, Berlin (1989) Mathematica Gottingensis
  • J P May, J Sigurdsson, Parametrized homotopy theory, Math. Surveys and Monogr. 132, Amer. Math. Soc. (2006)
  • B Norton-Odenthal, P Wong, A relative generalized Lefschetz number, Topology Appl. 56 (1994) 141–157
  • K Ponto, Equivariant fixed point theory
  • K Ponto, Fixed point theory and trace for bicategories, Astérisque (2010) xii+102
  • K Ponto, M Shulman, Shadows and traces in bicategories
  • K Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936) 586–593
  • H Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986) 459–473
  • H Schirmer, On the location of fixed points on pairs of spaces, Topology Appl. 30 (1988) 253–266
  • J Stallings, Centerless groups–-an algebraic formulation of Gottlieb's theorem, Topology 4 (1965) 129–134
  • A Strøm, Note on cofibrations, Math. Scand. 19 (1966) 11–14
  • F Wecken, Fixpunktklassen. II. Homotopieinvarianten der Fixpunkttheorie, Math. Ann. 118 (1941) 216–234
  • X Zhao, Minimal fixed point sets of relative maps, Fund. Math. 162 (1999) 163–180
  • X Zhao, On minimal fixed point numbers of relative maps, Topology Appl. 112 (2001) 41–70
  • X Zhao, Relative Nielsen theory, from: “Handbook of topological fixed point theory”, (R F Brown, M Furi, L Górniewicz, B J Jiang, editors), Springer, Dordrecht (2005) 659–684
  • X Zhao, A relative Reidemeister trace, JP J. Fixed Point Theory Appl. 1 (2006) 65–84