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


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 $$ N(f) = \frac{1}{[\pi_{1}(M):K]} \sum_{\bar\alpha \in \pi_{1}(M)/K} N(\bar\alpha\bar{f}).$$


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


Published: 2005
First available in Project Euclid: 16 August 2005

zbMATH: 1080.55003
MathSciNet: MR2145314

Primary: ‎55M20 , 57S30

Keywords: Infra-nilmanifolds , Lefschetz numbers , Nielsen numbers

Rights: Copyright © 2005 Editorial Board, Nagoya Mathematical Journal

Vol.178 • 2005
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