An Averaging Formula for Nielsen numbers on Infra-Solvmanifolds

TL;DR

This paper introduces a universal averaging formula for Nielsen numbers applicable to all self-maps on infra-solvmanifolds, extending previous specific cases and connecting to polynomial map Jacobians.

Contribution

It provides the first general formula for Nielsen numbers on all infra-solvmanifolds, unifying previous special cases and linking to polynomial map Jacobians.

Findings

The formula applies to all infra-solvmanifolds and self-maps.

It reduces to known formulas for special cases.

It relates Nielsen numbers to polynomial map Jacobians.

Abstract

Until now only for special classes of infra-solvmanifolds, namely infra-nilmanifolds and infra-solvmanifolds of type (R), there was a formula available for computing the Nielsen number of a self-map on those manifolds. In this paper, we provide a general averaging formula which works for all self-maps on all possible infra-solvmanifolds and which reduces to the old formulas in the case of infra-nilmanifolds or infra-solvmanifolds of type (R). Moreover, when viewing an infra-solvmanifold as a polynomial manifold, we recall that any map is homotopic to a polynomial map and we show how our formula can be translated in terms of the Jacobian of that polynomial map.