On a Theorem of Anosov on Nielsen Numbers for Nilmanifolds

Abstract

If f: M→M is a self-map of a compact manifold, it rarely happens that the Nielsen number n(f) is equal to the numerical value of the Lefschetz-Hopf number L(f), i.e., n(f) = |L.(f) ❘. For example, if M is an n-sphere, n ≥ 2, n(f) = 1, while L(f) = 1 + (−l)n deg f. On the positive side, n(f) = |L(f)❘ is valid for all tori but tori are the only compact Lie groups for which the result is valid [1]. In the early summer of 1984, D. V. Anosov (Steklov Mathematical Institute) wrote us inquiring whether it would be of interest to prove this result for compact nilmanifolds. Since nilmanifolds form a much larger class than Tori, we thought it worthwhile and wrote him this in a return letter. During the summer we worked out a proof of Anosov’s conjecture based on a product theorem for Nielsen numbers in fiber spaces [2] and sent it to Anosov. Later, we received a reply from Anosov, that he had worked out his proof in a purely geometric manner, deforming f to have exactly n(f) fixed points all having the same index 1 or −1.