# Commentarii Mathematici Helvetici

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**Volume 74, Issue 2, 1999, pp. 238–247**

**DOI: 10.1007/s000140050087**

Published online: 1999-06-30

Covering degrees are determined by graph manifolds involved

Shicheng Wang^{[1]}and F. Yu

^{[2]}(1) Peking University, Beijing, China

(2) Peking University, Beijing, China

W.Thurston raised the following question in 1976: Suppose that a compact 3-manifold M is not covered by (surface) $ \times S^1 $ or a torus bundle over $ S^1 $. If $ M_1 $ and $ M_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.

*Keywords: *Graph manifold, covering degree, B-matrix, H-matrix

Wang Shicheng, Yu F.: Covering degrees are determined by graph manifolds involved. *Comment. Math. Helv.* 74 (1999), 238-247. doi: 10.1007/s000140050087