Generalized Lefschetz numbers

Author:
S. Y. Husseini

Journal:
Trans. Amer. Math. Soc. **272** (1982), 247-274

MSC:
Primary 55M20

DOI:
https://doi.org/10.1090/S0002-9947-1982-0656489-X

MathSciNet review:
656489

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Abstract: Given , where is a finitely-generated -projective chain complex, and -chain map, with being a homomorphism, then the generalized Lefschetz number of is defined as the alternating sum of the -Reidemeister trace of . In analogy with the ordinary Lefschetz number, is shown to satisfy the commutative property and to be invariant under -chain homotopy. Also, when is -projective,

**[1]**Robert F. Brown,*The Lefschetz fixed point theorem*, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR**0283793****[2]**S. Eilenberg and N. E. Steenrod,*Foundations of algebraic topology*, Princeton Univ. Press, Princeton, N.J., 1951.**[3]**Edward Fadell and Sufian Husseini,*Fixed point theory for non-simply-connected manifolds*, Topology**20**(1981), no. 1, 53–92. MR**592570**, https://doi.org/10.1016/0040-9383(81)90014-8**[4]**D. H. Gottlieb,*A certain subgroup of the fundamental group*, Amer. J. Math.**87**(1965), 840–856. MR**0189027**, https://doi.org/10.2307/2373248**[5]**Dan McCord,*An estimate of the Nielsen number and an example concerning the Lefschetz fixed point theorem*, Pacific J. Math.**66**(1976), no. 1, 195–203. MR**0433443****[6]**John Stallings,*Centerless groups—an algebraic formulation of Gottlieb’s theorem*, Topology**4**(1965), 129–134. MR**0202807**, https://doi.org/10.1016/0040-9383(65)90060-1**[7]**Kurt Reidemeister,*Automorphismen von Homotopiekettenringen*, Math. Ann.**112**(1936), no. 1, 586–593 (German). MR**1513064**, https://doi.org/10.1007/BF01565432**[8]**Franz Wecken,*Fixpunktklassen. II. Homotopieinvarianten der Fixpunkttheorie*, Math. Ann.**118**(1941), 216–234 (German). MR**0010280**, https://doi.org/10.1007/BF01487362

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0656489-X

Article copyright:
© Copyright 1982
American Mathematical Society