Averaging formula for Nielsen coincidence numbers

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 $M$.