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The Anosov theorem for infranilmanifolds with an odd-order Abelian holonomy group. (English) Zbl 1113.55001

Given a selfmap \(f:M\to M\) on a compact manifold, the Nielsen number \(N(f)\) provides more information about the fixed point set of \(f\) than the Lefschetz number \(L(f)\) but the computation of \(N(f)\) is more difficult in general. If \(M\) is a nilmanifold, D. Anosov [Russ. Math. Surv. 40, No. 4, 149–150 (1985); translation from Usp. Mat. Nauk 40, No. 4 (244), 133–134 (1985; Zbl 0594.55002)] showed that \(N(f)=| L(f)| \) so the computation of \(N(f)\) reduces to that of the homological trace \(L(f)\). Anosov’s result has been generalized by many authors in different settings.
In the paper under review, the authors aim to investigate classes of manifolds for which the Anosov relation \(N(f)=| L(f)| \) holds for all selfmaps \(f\). If \(M\) is an infra-nilmanifold, \(E=\pi_1(M)\) fits in a short exact sequence \(1\to N\to E\to H \to 1\) where \(N\) is a torsion-free nilpotent group and the holonomy group \(H\) is finite. Using a result of K. B. Lee on semi-conjugation of affine endomorphism [Pac. J. Math. 168, No. 1, 157–166 (1995; Zbl 0920.55003)], the authors investigate the matrix calculation and show that the Anosov relation holds for all selfmaps if the holonomy group \(H\) is abelian of odd order. It is also pointed out that existing fiberwise techniques for solvmanifolds do not apply since \(E\) need not be strongly polycyclic or the fundamental group of a solvmanifold.

MSC:

55M20 Fixed points and coincidences in algebraic topology
53C29 Issues of holonomy in differential geometry
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References:

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