Nielsen numbers of periodic maps on solvmanifolds

Let f:M —► M be a self-map of a solvmanifold M. Then the Lefschetz number L(f) and the Nielsen number N(f) of / satisfy \L(f)\ 1 ), then !(/)=>(/). Let M be a closed manifold and let /: M —> M be a continuous map. We study the relations between the two invariants; the Lefschetz number L{f) and the Nielsen number N{f ). These two numbers give information on the existence of fixed point sets. If L{f) i1 0, every self-map g of M homotopic to / has a nonempty fixed point set. The Nielsen number is a lower bound for the number of components of the fixed point set of all maps homotopic to /. Even though N{f) gives more information than L{f) does, it is harder to calculate. For some cases, it is known that they are almost the same. If M is a torus, then \L{f)\ = N{f) [2]. If M is a nilmanifild, then \L{f)\ = N{f) [I]. If M is an infranilmanifold and / is homotopically periodic, then L{f) — Nif) [6]In fact, \L{f)\ — N{f) cannot be generalized to infranilmanifolds or solvmanifolds as [6, Theorem 10] shows. In [6] it was also shown that L{f) = N{f) cannot be generalized to nonperiodic maps. In this paper, we show L{f) = N{f) holds for solvmanifolds M if f has finite homotopy period. Let S be a connected, simply connected solvable Lie group and H be a closed subgroup of S. The coset space H\S is called a solvmanifold. We shall talk about compact solvmanifolds only. Theorem. Let f:M —> M be a self-map of a solvmanifold M. Then \L{f)\ < N{f) ■ If f ^ homotopically periodic then L{f) = N{f). The first part of the statement, which generalizes [1], has been proved in [9]. The second part is a generalization of certain cases of [6]. Note that there are Received by the editors March 20, 1991. 1991 Mathematics Subject Classification. Primary 55M20; Secondary 55M35.