Nielsen numbers of maps of tori
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- by Robin B. S. Brooks, Robert F. Brown, Jingyal Pak and Douglas H. Taylor PDF
- Proc. Amer. Math. Soc. 52 (1975), 398-400 Request permission
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.References
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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 & 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 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 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
Additional Information
- © Copyright 1975 American Mathematical Society
- 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