JournalscmhVol. 74 , No. 2DOI 10.1007/s000140050087

Covering degrees are determined by graph manifolds involved

  • Shicheng Wang

    Peking University, Beijing, China
  • F. Yu

    Peking University, Beijing, China
Covering degrees are determined by graph manifolds involved cover

Abstract

W.Thurston raised the following question in 1976: Suppose that a compact 3-manifold M is not covered by (surface) ×S1\times S^1 or a torus bundle over S1S^1 . If M1M_1 and M2M_2 are two homeomorphic finite covering spaces of M, do they have the same covering degree? For so called geometric 3-manifolds (a famous conjecture is that all compact orientable 3-manifolds are geometric), it is known that the answer is affirmative if M is not a non-trivial graph manifold. In this paper, we prove that the answer for non-trivial graph manifolds is also affirmative. Hence the answer for the Thurston's question is complete for geometric 3-manifolds. Some properties of 3-manifold groups are also derived.