Homeomorphisms of 3-manifolds and the realization of Nielsen number

Abstract

The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a map realizing the Nielsen number of f , which is a lower bound for the number of fixed points among all maps homotopic to f . The main theorem of this paper proves this conjecture for all orientation preserving maps on geometric or Haken 3-manifolds. It will also be shown that on many manifolds all maps are isotopic to fixed point free maps.The proof is based on the understanding of homeomorphisms on 2-orbifolds and 3-manifolds. Thurston's classification of surface homeomorphisms will be generalized to 2-dimensional orbifolds, which is used to study fiber preserving maps of Seifert fiber spaces. Maps on most Seifert fiber spaces are indeed isotopic to fiber preserving maps, with the exception of four manifolds and orientation reversing maps on lens spaces or S 3 . It will also be determined exactly which manifolds have a unique Seifert fibration up to isotopy. These informations will be used to deform a map to certain standard map on each piece of the JSJ decomposition, as well as on the neighborhood of the decomposition tori, which will make it possible to shrink each fixed point class to a single point, and remove inessential fixed point classes.

Twisted conjugacy classes in nilpotent groups

“…The isotopy Wecken theorem of Kelly also holds in dimension 2 [20] based upon Nielsen-Thurston classification of surface homeomorphisms and in dimension 3 [19] based upon techniques of 3-manidols. It is however unknown whether it holds in dimension 4 in general.…”

Dynamical zeta functions and asymptotic expansions in Nielsen theory

“…Proof The case of a surface with χ(F ) > 0 and case of torus were considered in [7]. If surface has χ(F ) = 0 and F is not a torus then any homeomorphism is isotopic to periodic one (see [11]) and Nielsen zeta function is a radical of rational function by lemma 1 . In the case of a hyperbolic (χ(F ) < 0) surface , according to Thurston's classification theorem, the homeomorphism f is isotopic either to a periodic or a pseudo-Anosov, or a reducible homeomorphism.…”

“…We use notation N(X) to denote a regular neighborhood of set X. We need some definitions from the paper [11]. [17], f /P is isotopic to a fiber preserving map.…”

The fixed subgroups of homeomorphisms of Seifert manifolds

“…A hyperbolic orbifold is orbifold covered by a hyperbolic surface and admits a hyperbolic structure with totally geodesic boundary. For more information about orbifolds, see [JWW,§1 and §2] and [S, §2]. A map f on a Seifert manifold M is called f iber-preserving if it maps fibers to fibers.…”