Any 3-manifold 1-dominates at most finitely many geometric 3-manifolds

Abstract

Maps between 3-manifolds has been studied by many people long times ago, and become an active subject again after Thurston's revolution on 3-manifold theory. We refer to [BW], [LWZ1] for various results and references on the subject.This paper addresses the following natural question which was raised around 1990, see also Kirby's Problem List,[K, 3.100].Question 1. Let M be a closed orientable 3-manifold. Are there at most finitely many closed, irreducible and orientable 3-manifolds N such that there exists a degree one map f : M → N ?Remarks on the conditions in Question 1. (i) If Poincare Conjecture fails, i.e., there is a homotopy 3-sphere N which is not S 3 , then one can get infinitely many reducible homotopy 3-spheres by doing connected sums on N . Since there always exists degree one map from a 3-manifold M to a homotopy 3-sphere, the condition "irreducible" on the target N is posed to avoid this unclear case.(ii) The condition "closed" is posed on M and N just for simplicity. Indeed we can replace "closed" by "compact", and meanwhile replace "degree one map" by "degree one proper map". A map f :For simplicity, we adapt the following definition from [BW]. Let M and N be two compact orientable 3-manifolds. Say M (1-)dominates N if there is a proper map f : M → N of non-zero degree (degree 1).A closed orientable 3-manifold is called geometric if it admits one of the following geometries: H 3 (hyperbolic),

Roots of torsion polynomials and dominations

“…It holds for dominations of hyperbolic manifolds by Soma [6], for H 2 × R manifolds by Wang and Zhou [9], and for Sol manifolds by Corollary 3.6.…”

“…Corollary 3.3 shows that there are only finitely many possibilities for trace(ϕ), which is the negative of the coefficient of t in ∆ T 2 ×I ϕ (t). On the other hand, there are only finitely many SL 2 (Z) conjugacy classes of hyperbolic elements of SL 2 (Z) with a given trace (eg see Wang and Zhou [9,Lemma 8]). Since the homeomorphism type of a torus bundle over the circle depends only on the conjugacy class of its monodromy ϕ ∈ SL 2 (Z), it follows that a closed, connected, orientable 3-manifold can dominate at most finitely many torus bundles over the circle with hyperbolic monodromy.…”

Local rigidity of aspherical three-manifolds

“…This comes from the following observation: by [25,Lemma 6] there are rational numbers r i , s i and a vertical surface W i in P i (i.e. an incompressible properly embedded surface in P i which is fibered by the S 1 -fibers of P i ) such that…”

Degree-one maps between hyperbolic 3-manifolds with the same volume limit

“…Though this stronger rigidity theorem is powerful, one cannot apply it directly to some of the subjects which are important in the study of degree-one maps between geometric 3-manifolds. We refer to [16], [1], [6], [7], [24] and their references for various results and related topics concerning such subjects.…”