| The 3-manifold S1 × S2 is a prime one, but has essential 2-spheres. It is an important 3-manifold that was studied much more. In this paper, we consider the group structure of mapping class group of connected sum of S1 × S2. Prom these studying, we can understand more about the connected sum of S1 × S2 and their topologies.The mapping class group of S1 × S2 is Z2 (?) Z2 has already been obtained. J. H. Rubinstein have obtained the mapping class groups of the closed, irreducible and orientable 3-manifolds which contain Klein bottles and have finite fundamental groups. And for the reducible 3-manifolds, Darryl McCullough have proved that: if M is a compact connected orientable 3-manifold, then any orientation-preserving homeomorphism of M is isotopic to a composite of the following four types of homeomorphisms: homeomorphisms preserving summands, interchanges of homeomorphic summands, spins of S1 × S2 summands, slide homeomorphisms. In this paper, we will give a calculation to the mapping class group of S1 × S2#S1 × S2 according to Darryl McCullough's theorem. We do the further study on these four types of homeomorphisms in the case of S1 × S2#S1 × S2, look for some simple generators, and try to give the relation and the structure of mapping class group of S1 × S2#S1 × S2. Finally, we present the group structure of mapping class subgroups of generated by the first three types homeomorphisms. |
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