Covering Invariants and Cohopficity of 3-manifold Groups

A 3-manifold M is called to have property C if the degrees of finite coverings over M are determined by the manifolds. Thurston asked the problems of whether there is a real valued invariant of 3-manifolds with a certain covering property, and what 3-manifolds have property C. For geometric manifolds the conjecture is that M has property C if and only if it is not covered by surface×S or torus bundle over S. Using the Gromov norm, the conjecture is quickly reduced to the case of graph manifolds. In this paper we define a new invariant ω(M) for graph manifolds. It has the covering property that if N is a d-fold covering space of M , then ω(N) = dω(M). Thus in the generic case, when ω(M) 6= 0, the manifold M has property C. We also use the methods to prove two other results. The first classifies all geometric manifolds that admit nontrivial self coverings. The second solves a problem of Gonzales-Acuna and Whitten about the cohopficity of 3-manifold groups, and determines all closed geometric manifolds that have cohopfian fundamental groups. Definition. A 3-manifold M is called to have property C if, whenever M1,M2 are homeomorphic covering spaces of M , the degrees of the coverings are the same. It has property Cr if the above is true for all regular coverings. We are interested in the following problems of Thurston: (1) [7, Problem 3.16] Is there a reasonable real valued invariant C(M) such that (a) if M = M1#M2 then C(M) = C(M1) + C(M2); (b) the covering property holds, i.e, if M1 is a k-fold covering of M , then C(M1) = kC(M)? C(M) may have some additional properties. (2) [7, Problem 3.16(a)] What 3-manifolds have property C? An invariant of 3-manifolds with the covering property will give a partial solution to the Property C problem: if C(M) 6= 0, then the covering degree of M1 → M is given by k = C(M1)/C(M), so M has property C. There are some well known covering invariants. The simplest one is the number of 2-spheres on the boundary of an orientable manifold, and the most interesting one is the Gromov norm ν(M) defined by Gromov [3] soon after these questions were posed. The following definition is convenient for our purpose. Mathematics Subject Classification: 57M10, 57N10 Partially supported by NSF of China Partially supported by NSF grant DMS 9102633 1 Definition. A compact, connected, orientable 3-manifold M is called geometric, if M is either a Seifert manifold, or a hyperbolic manifold, or a Haken manifold, or a connected sum of such manifolds. For a closed geometric 3-manifold M , ν(M) is essentially the sum of the volumes of all hyperbolic pieces in the geometric decomposition of M ; therefore M has property C if it contains some hyperbolic pieces. Note that the Gromov norm is a zero function on the set of all closed 3-manifolds which contain no hyperbolic pieces. We only need to consider orientable manifolds, for if M is a nonorientable 3-manifold, then it has property C if and only if its orientable double covering does. So we make the following Convention. Except in section 8, all 3-manifolds in this paper are assumed orientable. A famous conjecture is that all compact 3-manifolds are geometric. Also, it is known that if M is covered by (surface)×S1 or torus bundle over S1, then it does not have property C. Actually such manifolds admit nontrivial self coverings (see Theorem 8.6). Therefore we will concentrate on 3-manifolds in the class G defined below. Notation. We use G to denote the set of all geometric 3-manifolds which are not covered by either (surface)×S1 or torus bundle over S1. Conjecture 1: Every 3-manifold M in G has Property C. A geometric 3-manifold M is called a graph manifold if it has a nonempty canonical splitting surface (see section 1) cutting M into Seifert manifold pieces. It can be shown (Thm 2.5) that Conjecture 1 is true if M is not a graph manifold, so the problem is reduced to whether graph manifolds have property C. The main result of this paper is to define a new invariant ω(M) for graph manifolds, which takes values in R. It is defined in terms of the Euler characteristics of orbifolds of the Seifert fibered pieces of M , and the intersection numbers of fibers on splitting surfaces. The central feature of ω(M) is the covering property. Hence in the generic case, when ω(M) is nonzero, M has property C. A closely related problem is the cohopficity of 3-manifold groups. A group is called cohopfian if it is not isomorphic to a proper subgroup of itself. In [1] Gonzales-Acuna and Whitten studied this problem, and determined the cohopficity of the fundamental groups