Estimating Nielsen Numbers on Infrasolvmanifolds

A well-known lower bound for the number of xed points of a self-map f : X ?! X is the Nielsen number N(f). Unfortunately, the Nielsen number is diicult 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 xed points. In this paper, we investigate conditions on the space X which guarantee either N(f) = jL(f)j or N(f) jL(f)j. By considering the Nielsen and Lefschetz coincidence numbers, we show that N(f) jL(f)j for all self-maps on compact infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable group of nite index). Moreover, for infranilmanifolds, there is a Lefschetz number formula which computes N(f). Consider a continuous self map f : X ?! X. Let Fix(f) denote the xed point set fx 2 X j f(x) = xg. One of the fundamental problems of xed 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 xed points of g, for all maps g homotopic to f. This estimate is sharp for all compact manifolds save surfaces of negative Euler characteristic. Its one drawback is that it is very diicult to compute N(f) from its deenition, so that other means must be sought. At least, since the Nielsen number provides a lower bound for the original topological object jFix(f)j, it would be useful to nd 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 algbraic-topological means of nding 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 jFix(f)j. One approach to computing the Nielsen number is to nd conditions on either the space X or the map f which allow N(f) and L(f) to be related. The Jiang condition, for example, is a condition on the map f which, when satissed, computes N(f) from L(f) and coker(1 ? f 1). The other approach, searching for conditions on the space X, begins with the result of Brooks, Brown, Pak and Taylor 8], that N(f) = jL(f)j for …