Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On averaging Lefschetz numbers


Author: U. Kurt Scholz
Journal: Proc. Amer. Math. Soc. 33 (1972), 607-612
MSC: Primary 55C20
MathSciNet review: 0343262
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let (E, p, X) be a regular covering space where E is a connected metric ANR (absolute neighborhood retract) and let $ f:X \to X$ be a map. This paper investigates the relationship between the Lefschetz number of f and those of its lifts, i.e. maps $ f':E \to E$ so that $ pf' = fp$. In particular, it is shown that to a lift $ f':E \to E$ one may associate a class of lifts $ \mathfrak{L}(f')$ with the property that the Lefschetz number of f is equal to the average of the Lefschetz numbers of maps in $ \mathfrak{L}(f')$.


References [Enhancements On Off] (What's this?)

  • [1] Felix E. Browder, Asymptotic fixed point theorems, Math. Ann. 185 (1970), 38–60. MR 0275408
  • [2] Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR 0283793
  • [3] A. Granas, The Hopf-Lefschetz fixed point theorem for noncompact ANR's, Proc. Conference on Infinite Dimensional Topology, Baton Rouge, La., 1967.
  • [4] Olof Hanner, Some theorems on absolute neighborhood retracts, Ark. Mat. 1 (1951), 389–408. MR 0043459
  • [5] P. J. Hilton and S. Wylie, Homology theory: An introduction to algebraic topology, Cambridge University Press, New York, 1960. MR 0115161
  • [6] Heinz Hopf, Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), no. 1, 493–524 (German). MR 1545024, 10.1007/BF01180550
  • [7] Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
  • [8] Bo-ju Jiang, Estimation of the Nielsen numbers, Chinese Math. – Acta 5 (1964), 330–339. MR 0171279
  • [9] D. McCord, The converse of the Lefschetz fixed point theorem for surfaces and higher dimensional manifolds, Doctoral Dissertation, University of Wisconsin, Madison, Wis., 1970.
  • [10] U. K. Scholz, The Nielsen theory on fixed point classes for Palais maps, Doctoral Dissertation, University of California, Los Angeles, Calif., 1970.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55C20

Retrieve articles in all journals with MSC: 55C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0343262-4
Keywords: Lefschetz number, regular covering space, fixed point index
Article copyright: © Copyright 1972 American Mathematical Society