Abstract
The idea of the proof is to investigate when a homology cobordism between closed 4-rnanifolds with infinite cyclic fundamental groups is an h-cobordism which is always a product cobordism by Freedman [2], for we can construct a homology cobordism between and S x *S3fl:M1 by a method similar to Kervaire's surgery argument [11]. Freedman showed in [1] that any two closed oriented simply connected 4-manifolods Mly M are orientation-preservingly homeomorphic if and only if the intersection forms on H2(M1'yZ), H2(M\\Z) are isomorphic and the KirbySiebenmann invariants ks(7W\), ks(7kf'1)(eZ2) are equal, and in this case there is an orientation-preserving homeomorphism M±~M inducing the isomorphism of the intersection forms. By combining this classification of Freedman with the above splitting theorem, we have a similar characterization for closed oriented 4-manifolds with infinite cyclic fundamental groups:
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