Nielsen numbers of maps of tori

Brooks, Robin B. S.; Brown, Robert F.; Pak, Jingyal; Taylor, Douglas H.

doi:10.1090/S0002-9939-1975-0375287-X

Nielsen numbers of maps of tori
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by Robin B. S. Brooks, Robert F. Brown, Jingyal Pak and Douglas H. Taylor

Proc. Amer. Math. Soc. 52 (1975), 398-400

DOI: https://doi.org/10.1090/S0002-9939-1975-0375287-X

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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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