Algebraic & Geometric Topology

The multiplicativity of fixed point invariants

Kate Ponto and Michael Shulman

Full-text: Open access

Abstract

We prove two general factorization theorems for fixed-point invariants of fibrations: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar multiplicativity results for the Lefschetz and Nielsen numbers of a fibration. Moreover, the proofs of these theorems are essentially formal, taking place in the abstract context of bicategorical traces. This makes generalizations to other contexts straightforward.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 3 (2014), 1275-1306.

Dates
Received: 5 January 2013
Revised: 27 October 2013
Accepted: 28 October 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715904

Digital Object Identifier
doi:10.2140/agt.2014.14.1275

Mathematical Reviews number (MathSciNet)
MR3190594

Zentralblatt MATH identifier
1293.55003

Subjects
Primary: 55M20: Fixed points and coincidences [See also 54H25]
Secondary: 18D05: Double categories, 2-categories, bicategories and generalizations 55R05: Fiber spaces

Keywords
Lefschetz number Reidemeister trace Nielsen number trace

Citation

Ponto, Kate; Shulman, Michael. The multiplicativity of fixed point invariants. Algebr. Geom. Topol. 14 (2014), no. 3, 1275--1306. doi:10.2140/agt.2014.14.1275. https://projecteuclid.org/euclid.agt/1513715904


Export citation

References

  • M F Atiyah, Thom complexes, Proceedings London Math. Soc. 11 (1961) 291–310
  • R F Brown, The Nielsen number of a fibre map, Ann. of Math. 85 (1967) 483–493
  • R F Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co, Glenview, Il (1971)
  • R F Brown, E R Fadell, Corrections to: “The Nielsen number of a fibre map”, Ann. of Math. 95 (1972) 365–367
  • M G Citterio, The Reidemeister number as a homotopy equalizer, Rend. Mat. Appl. 18 (1998) 87–101
  • S R Costenoble, S Waner, Equivariant ordinary homology and cohomology
  • A Dold, D Puppe, Duality, trace, and transfer, from: “Proceedings of the International Conference on Geometric Topology”, (K Borsuk, editor), PWN, Warsaw (1980) 81–102
  • P R Heath, Fibre techniques in Nielsen theory calculations, from: “Handbook of topological fixed point theory”, (R F Brown, M Furi, L Górniewicz, B Jiang, editors), Springer, Dordrecht (2005) 489–554
  • P R Heath, E Keppelmann, P N-S Wong, Addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps, Topology Appl. 67 (1995) 133–157
  • P R Heath, C Morgan, R Piccinini, Nielsen numbers and pullbacks, Topology Appl. 26 (1987) 65–82
  • S Y Husseini, Generalized Lefschetz numbers, Trans. Amer. Math. Soc. 272 (1982) 247–274
  • B J Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 14, Amer. Math. Soc. (1983)
  • L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer, Berlin (1986)
  • J P May, J Sigurdsson, Parametrized homotopy theory, Mathematical Surveys and Monographs 132, Amer. Math. Soc. (2006)
  • B E Norton-Odenthal, A product formula of the generalized Lefschetz number, PhD thesis, The University of Wisconsin-Madison (1991)
  • J Pak, On the fixed point indices and Nielsen numbers of fiber maps on Jiang spaces, Trans. Amer. Math. Soc. 212 (1975) 403–415
  • K Ponto, Coincidence invariants and higher Reidemeister traces
  • K Ponto, Fixed point theory and trace for bicategories, Astérisque 333, Soc. Math. France, Paris (2010)
  • K Ponto, M Shulman, Traces in symmetric monoidal categories, to appear in Expositiones Mathematicae Available at \setbox0\makeatletter\@url http://www.sciencedirect.com/science/article/pii/S0723086913000790 {\unhbox0
  • K Ponto, M Shulman, Duality and traces for indexed monoidal categories, Theory Appl. Categ. 26 (2012) 582–659
  • K Ponto, M Shulman, Shadows and traces in bicategories, J. Homotopy Relat. Struct. 8 (2013) 151–200
  • M Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008) No. 18, 650–738
  • C Y You, Fixed point classes of a fiber map, Pacific J. Math. 100 (1982) 217–241