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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1240025-7

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 is the Nielsen number . Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number , 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), when is a homotopically periodic map.

**[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)**

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1240025-7

Article copyright:
© Copyright 1994
American Mathematical Society