The Anosov theorem for infranilmanifolds with cyclic holonomy group

Karel Dekimpe, Bram De Rock and Wim Malfait

Vol. 229 (2007), No. 1, 137–160

DOI: 10.2140/pjm.2007.229.137

Abstract

A celebrated theorem of Anosov states that for any continuous self-map f : M → M of a nilmanifold M, the Nielsen number equals the Lefschetz number in absolute value. Anosov also showed that this result does not hold for infranilmanifolds, even in the simplest possible situation of flat manifolds with cyclic holonomy group.

Nevertheless, in this paper we extend Anosov’s theorem to infranilmanifolds with cyclic holonomy group, provided a certain easily checked condition on the holonomy representation is satisfied.

In the case of flat manifolds with cyclic holonomy group this condition is necessary and sufficient. In the general case of all infranilmanifolds with cyclic holonomy group, we provide an example which shows that this condition is no longer necessary.

We also prove that for any nonorientable flat manifold Anosov’s theorem is not true, but again the same example shows that this is not valid in general for nonorientable infranilmanifolds.

Keywords

Nielsen number, Lefschetz number, infranilmanifolds, fixed point theory

Mathematical Subject Classification 2000

Primary: 55M20, 37C25

Milestones

Received: 2 May 2005

Accepted: 28 November 2005

Published: 1 January 2007