Nagoya Mathematical Journal

Averaging formula for Nielsen numbers

Seung Won Kim, Jong Bum Lee, and Kyung Bai Lee

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We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-nilmanifolds: Let $M$ be an infra-nilmanifold and $f : M \to M$ be a continuous map. Suppose $M_{K}$ is a regular covering of $M$ which is a compact nilmanifold with $\pi_{1}(M_{K}) = K$. Assume that $f_{*}(K) \subset K$. Then $f$ has a lifting $\bar{f} : M_{K} \to M_{K}$ on $M_{K}$. We prove a question raised by McCord, which is for an $\alpha \in \pi_{1}(M)$ with $p$(Fix$(\alpha\tilde{f}))$ an essential fixed point class, Fix$(\tau_{\alpha}\varphi) = 1$. As a consequence, we obtain the following averaging formula for Nielsen numbers $$ N(f) = \frac{1}{[\pi_{1}(M):K]} \sum_{\bar\alpha \in \pi_{1}(M)/K} N(\bar\alpha\bar{f}).$$

Article information

Nagoya Math. J., Volume 178 (2005), 37-53.

First available in Project Euclid: 16 August 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25] 57S30: Discontinuous groups of transformations

Infra-nilmanifolds Lefschetz numbers Nielsen numbers


Kim, Seung Won; Lee, Jong Bum; Lee, Kyung Bai. Averaging formula for Nielsen numbers. Nagoya Math. J. 178 (2005), 37--53.

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