Estimating Nielsen numbers on infrasolvmanifolds

Abstract

A well-known lower bound for the number of fixed points of a selfmap /: X —» X is the Nielsen number N(f). Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number L(f), on the other hand, is readily computable, but does not give a lower bound for the number of fixed points. In this paper, we investigate conditions on the space X which guarantee either N(f) = \L(f) or N(f) > \L(f) . By considering the Nielsen and Lefschetz coincidence numbers, we show that N(f) > \L(f) for all self-maps on compact infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable subgroup of finite index). Moreover, for infranilmanifolds, there is a Lefschetz number formula which computes N(f). 1. Estimating Nielsen numbers. Consider a continuous self-map /: X -> X. Let Fix(/) denote the fixed point set {x e X\f(x) = x} . One of the fundamental problems of fixed point theory is to estimate (preferably from below) the cardinality of this set. The Nielsen number N(f) provides such an estimate: it is an integer homotopy invariant which provides a lower bound on the number of fixed points of g, for all maps g homotopic to /. This estimate is sharp for all compact manifolds save surfaces of negative Euler characteristic. Its one drawback is that it is very difficult to compute N(f) from its definition, so that other means must be sought. At least, since the Nielsen number provides a lower bound for the original topological object |Fix(/)|, it would be useful to find lower bounds for N(f). We will refer to the search for lower bounds to N(f) as the problem of estimating N(f) while the search for other algebraic-topological means of finding the exact value of N(f) will be referred to as the problem of computing N(f). The Lefschetz number L(f) is a (reasonably) computable invariant, but in general, there is no relation between L(f) and either N(f) or |Fix(/)|. One approach to computing the Nielsen number is to find conditions on either the space X or the map / which allow N(f) and L(f) to be related. The Jiang condition, for example, is a condition on the map / which, when satisfied, computes N(f) from L(f)