JournalscmhVol. 74, No. 2pp. 238–247

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
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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.

Cite this article

Shicheng Wang, F. Yu, Covering degrees are determined by graph manifolds involved. Comment. Math. Helv. 74 (1999), no. 2, pp. 238–247

DOI 10.1007/S000140050087