Charlap in 1967 gave the first example of non-homeomorphic flat manifolds M and N such that S1×M is homeomorphic to S1×N, where M, N are of dimension greater than 37. In 1972, Conner and Raymond gave a sufficient condition for closed orientable 3-manifolds M3, N3 with S1×M3 is diffeomorphic to S1×N3. In 1993, Shy completed the above classification problem by giving the necessary condition. In this paper, we show that closed orientable 3-manifolds M3 and N3 diffeomorphic to S1×M3 diffeomorphic to S1×N3 admit the property that M3 and N3 may cover each other. More precisely, M3 is a k-fold covering of N3, and N3 is an m-fold covering of M3.
|Number of pages||5|
|Journal||Proceedings of the National Science Council, Republic of China, Part A: Physical Science and Engineering|
|Publication status||Published - 1997 Nov 1|
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