×

The multiplicativity of fixed point invariants. (English) Zbl 1293.55003

Given a fibration \(p: E\to B\), a fibre-preserving self-map \(f: E \to E\) induces a self-map \(\bar f\) on the base space \(B\). A fixed point \(b\) of \(\bar f\) gives a self-map \(f_b: F_b\to F_b\) on the fibre \(F_b=p^{-1}(b)\). P. Heath et al. [Topology Appl. 26, 65–82 (1987; Zbl 0618.55002)] obtained a product formula for the Lefschetz number, namely \(L(f) = L(\bar f) L(f_b)\), if \(\pi_1(B)\) acts trivially on the fibres. The authors of the paper under review obtain a generalization and factorization of the above product formula by using the notion of bicategorical trace introduced by themselves. Here, a more delicate invariant, the Reidemeister trace, is in consideration. All invariants are regarded as homomorphisms between abelian groups, and hence a product of invariants will turn out to be a composition of homomorphisms. Since the Reidemeister trace includes the information of Lefschetz number, Nielsen number and fixed point indices, their factorization implies most of the existing product formulas for those invariants. Moreover, the authors simplify and unify the additional conditions imposed on the fibration to guarantee those product formulae.

MSC:

55M20 Fixed points and coincidences in algebraic topology
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
55R05 Fiber spaces in algebraic topology

Citations:

Zbl 0618.55002
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M F Atiyah, Thom complexes, Proceedings London Math. Soc. 11 (1961) 291 · Zbl 0124.16301 · doi:10.1112/plms/s3-11.1.291
[2] R F Brown, The Nielsen number of a fibre map, Ann. of Math. 85 (1967) 483 · Zbl 0149.20304 · doi:10.2307/1970354
[3] R F Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co (1971) · Zbl 0216.19601
[4] R F Brown, E R Fadell, Corrections to: “The Nielsen number of a fibre map”, Ann. of Math. 95 (1972) 365 · Zbl 0234.55019 · doi:10.2307/1970803
[5] M G Citterio, The Reidemeister number as a homotopy equalizer, Rend. Mat. Appl. 18 (1998) 87 · Zbl 0903.55002
[6] S R Costenoble, S Waner, Equivariant ordinary homology and cohomology, · Zbl 1362.55001
[7] A Dold, D Puppe, Duality, trace, and transfer (editor K Borsuk), PWN (1980) 81 · Zbl 0473.55008
[8] P R Heath, Fibre techniques in Nielsen theory calculations (editors R F Brown, M Furi, L Górniewicz, B Jiang), Springer (2005) 489 · Zbl 1077.55002 · doi:10.1007/1-4020-3222-6_14
[9] 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 · Zbl 0845.55004 · doi:10.1016/0166-8641(95)00019-8
[10] P R Heath, C Morgan, R Piccinini, Nielsen numbers and pullbacks, Topology Appl. 26 (1987) 65 · Zbl 0618.55002 · doi:10.1016/0166-8641(87)90026-5
[11] S Y Husseini, Generalized Lefschetz numbers, Trans. Amer. Math. Soc. 272 (1982) 247 · Zbl 0507.55001 · doi:10.2307/1998959
[12] B J Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 14, Amer. Math. Soc. (1983) · Zbl 0512.55003 · doi:10.1090/conm/014
[13] L G Lewis Jr., J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer (1986) · Zbl 0611.55001
[14] J P May, J Sigurdsson, Parametrized homotopy theory, Mathematical Surveys and Monographs 132, Amer. Math. Soc. (2006) · Zbl 1119.55001
[15] B E Norton-Odenthal, A product formula of the generalized Lefschetz number, PhD thesis, The University of Wisconsin-Madison (1991)
[16] J Pak, On the fixed point indices and Nielsen numbers of fiber maps on Jiang spaces, Trans. Amer. Math. Soc. 212 (1975) 403 · Zbl 0308.55003 · doi:10.2307/1998637
[17] K Ponto, Coincidence invariants and higher Reidemeister traces, · Zbl 1342.18010
[18] K Ponto, Fixed point theory and trace for bicategories, Astérisque 333, Soc. Math. France (2010) · Zbl 1207.18001
[19] K Ponto, M Shulman, Traces in symmetric monoidal categories, to appear in Expositiones Mathematicae · Zbl 1308.18008 · doi:10.1016/j.exmath.2013.12.003
[20] K Ponto, M Shulman, Duality and traces for indexed monoidal categories, Theory Appl. Categ. 26 (2012) 582 · Zbl 1275.18019
[21] K Ponto, M Shulman, Shadows and traces in bicategories, J. Homotopy Relat. Struct. 8 (2013) 151 · Zbl 1298.18006 · doi:10.1007/s40062-012-0017-0
[22] M Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008) 650 · Zbl 1192.18005
[23] C Y You, Fixed point classes of a fiber map, Pacific J. Math. 100 (1982) 217 · Zbl 0512.55004 · doi:10.2140/pjm.1982.100.217
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.