Relative Euler Number and Finite Covers of Graph Manifolds

In this paper we show that every non-trivial graph manifold M has a finite cover that contains a foliation which is very close to a fibration over the circle — the foliation restricted to each vertex manifold of M is a fibration over the circle. On the other hand, we show that this foliation can not always be made to be a fibration over the circle, by exhibiting a large class of graph manifolds that have no finite cover that fibers over the circle. These computations are done using a “relative Euler number” for framed Seifert fibered spaces that are assigned to the vertex manifolds of M . This relative Euler number behaves naturally under finite covers. Using the relative Euler number we define a torsion for each vertex manifold of M , and show that if the torsions are all positive then M is not a surface bundle over S. Using the naturality of Euler number with respect to covers, we show that under the same conditions in fact no finite cover of M will fiber over the circle. This situation is analogous to that of Seifert fibered spaces. It follows from [EHN] that every Seifert fibered space with base orbifold of negative characteristic is covered by a manifold admitting a horizontal foliation. On the other hand a Seifert fibered space is covered by a surface bundle over S if and only if its Euler number or its orbifold characteristic is zero. We also use this relative Euler number to define a covering invariant σ(M) (in the sense of [WW] and [K]). In particular it follows that if σ(M) is non-zero then M will have the property that homeomorphic covers of M will have the same covering degree. It seems rare that σ(M) is zero. For example, we show that if M is a knot complement in the 3-sphere then σ(M) is non-zero. We refer the reader to [J,H,Sc] for definitions and basic facts about Seifert fibered spaces, and to [Sc] for the theory of 2-dimensional orbifolds, especially their Euler characteristic. If N is a Seifert fibered space, we use O(N) to denote its orbifold, and use χ(O(N)) to denote the Euler characteristic of its orbifold, called the orbifold characteristic of N . After Waldhausen [W, page 87], an orientable, connected, compact 3-manifold is a graph manifold if and only if there is a collection of tori T = T1 ∪ . . . ∪ Tk in the interior of M such that the each component of M − IntN(T ) is a circle