Nagoya Mathematical Journal

Averaging formula for Nielsen coincidence numbers

Seung Won Kim and Jong Bum Lee

Full-text: Open access

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 $M$.

Article information

Source
Nagoya Math. J., Volume 186 (2007), 69-93.

Dates
First available in Project Euclid: 22 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1182525230

Mathematical Reviews number (MathSciNet)
MR2334365

Zentralblatt MATH identifier
1129.55002

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

Keywords
Infra-nilmanifold Lefschetz coincidence number Nielsen coincidence number Semi-index

Citation

Kim, Seung Won; Lee, Jong Bum. Averaging formula for Nielsen coincidence numbers. Nagoya Math. J. 186 (2007), 69--93. https://projecteuclid.org/euclid.nmj/1182525230


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