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2007 Averaging formula for Nielsen coincidence numbers
Seung Won Kim, Jong Bum Lee
Nagoya Math. J. 186: 69-93 (2007).

Abstract

In this paper we study the averaging formula for Nielsen coincidence numbers of pairs of maps $(f, g) : M \to N$ between closed smooth manifolds of the same dimension. Suppose that $G$ is a normal subgroup of $\Pi = \pi_{1}(M)$ with finite index and $H$ is a normal subgroup of $\Delta = \pi_{1}(N)$ with finite index such that $f_{*}(G) \subset H$ and $g_{*}(G) \subset H$. Then we investigate the conditions for which the following averaging formula holds

N(f, g) = \frac{1}{[\Pi:G]} \sum_{\bar{\alpha} \in \Delta/H} N(\bar{\alpha} \bar{f}, \bar{g}),

where $(\bar{f}, \bar{g}) : M_{G} \to N_{H}$ is any pair of fixed liftings of $(f, g)$. We prove that the averaging formula holds when $M$ and $N$ are orientable infra-nilmanifolds of the same dimension, and when $M = N$ is a non-orientable infra-nilmanifold with holonomy group $\mathbb{Z}_{2}$ and $(f, g)$ admits a pair of liftings $(\bar{f}, \bar{g}) : \bar{M} \to \bar{M}$ on the nil-covering $\bar{M}$ of $M$.

Citation

Download Citation

Seung Won Kim. Jong Bum Lee. "Averaging formula for Nielsen coincidence numbers." Nagoya Math. J. 186 69 - 93, 2007.

Information

Published: 2007
First available in Project Euclid: 22 June 2007

zbMATH: 1129.55002
MathSciNet: MR2334365

Subjects:
Primary: ‎55M20, 57S30

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal

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