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)



Nielsen numbers of homotopically periodic maps on infrasolvmanifolds

Author: Christopher K. McCord
Journal: Proc. Amer. Math. Soc. 120 (1994), 311-315
MSC: Primary 55M20; Secondary 57N65
MathSciNet review: 1240025
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A well-known lower bound for the number of fixed points of a self-map $ f:X \to X$ is the Nielsen number $ N(f)$. Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number $ L(f)$, on the other hand, is readily computable but usually does not estimate the number of fixed points. In this paper, we show that on infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable group of finite index), $ N(f) = L(f)$ when $ f$ is a homotopically periodic map.

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

  • [1] D. V. Anosov, The Nielsen number of maps of nil-manifolds, Russian Math. Surveys 40 (1985), 149-150.
  • [2] L. Auslander and F. E. A. Johnson, On a conjecture of C. T. C. Wall, J. London Math. Soc. (2) 14 (1976), 331-332. MR 0423362 (54:11341)
  • [3] R. Brooks, R. Brown, J. Pak, and D. Taylor, Nielsen numbers of maps of tori, Proc. Amer. Math. Soc. 52 (1975), 398-400. MR 0375287 (51:11483)
  • [4] E. Fadell and S. Husseini, On a theorem of Anosov on Nielsen numbers for nilmanifolds, Nonlinear Functional Analysis and its Applications (Maratea, 1985), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 173, Reidel, Dordrecht and Boston, MA, 1986, pp. 47-53. MR 852569 (87j:55004)
  • [5] F. T. Farrell and L. E. Jones, Classical aspherical manifolds, CBMS Lecture Notes, vol. 75, Amer. Math. Soc., Providence, RI, 1990. MR 1056079 (91k:57001)
  • [6] S. Kwasik and K. B. Lee, The Nielsen numbers of homotopically periodic maps of infranilmanifolds, J. London Math. Soc. (2) 38 (1988), 544-554. MR 972137 (90b:55002)
  • [7] K. B. Lee, Nielsen numbers of periodic maps on solvmanifolds, Proc. Amer. Math. Soc. 116 (1992), 575-579. MR 1123659 (92m:55002)
  • [8] C. McCord, Nielsen numbers and Lefschetz numbers on solvmanifolds, Pacific J. Math. 147 (1991), 153-164. MR 1081679 (92a:55002)
  • [9] C. McCord, Estimating Nielsen numbers on infrasolvmanifolds, Pacific J. Math. 154 (1992), 345-368. MR 1159516 (93d:55001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55M20, 57N65

Retrieve articles in all journals with MSC: 55M20, 57N65

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society