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.