Nielsen numbers of maps of tori
Authors:
Robin B. S. Brooks, Robert F. Brown, Jingyal Pak and Douglas H. Taylor
Journal:
Proc. Amer. Math. Soc. 52 (1975), 398-400
MSC:
Primary 55C20
DOI:
https://doi.org/10.1090/S0002-9939-1975-0375287-X
MathSciNet review:
0375287
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Abstract | References | Similar Articles | Additional Information
Abstract: The main result states that if $f:X \to X$ is any map on a $k$-dimensional torus $X$, then the Nielsen number and Lefschetz number of $f$ are related by the formula $N(f) = |L(f)|$. Thus, on the torus, the Lefschetz number gives information, not just on the existence of fixed points, but on the number of fixed points as well. No other compact Lie group has this property. The main result, when applied to certain types of maps on compact Lie groups, produces new information on the fixed point theory of such maps.
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R. Brooks, Coincidences, roots and fixed points, Doctoral Dissertation, University of California, Los Angeles, Calif., 1967.
- Robert F. Brown, Fixed points and fibre, Pacific J. Math. 21 (1967), 465–472. MR 214069
- Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR 0283793
- Robert F. Brown and Edward R. Fadell, Corrections to: “The Nielsen number of a fibre map”, Ann. of Math. (2) 95 (1972), 365–367. MR 317325, DOI https://doi.org/10.2307/1970803
- Edward Fadell, Recent results in the fixed point theory of continuous maps, Bull. Amer. Math. Soc. 76 (1970), 10–29. MR 271935, DOI https://doi.org/10.1090/S0002-9904-1970-12358-8 ---, Review #4921, Math. Rev. 35 (1968), 909.
- Bo-ju Jiang, Estimation of the Nielsen numbers, Chinese Math. – Acta 5 (1964), 330–339. MR 0171279
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Additional Information
Keywords:
Nielsen number,
Lefschetz number,
fixed point,
compact Lie group,
homogeneous space
Article copyright:
© Copyright 1975
American Mathematical Society
