Splitting a 4-manifold with infinite cyclic fundamental group
Splitting a 4-manifold with infinite cyclic fundamental group
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DOI：
10.18910/3812
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发表时间：
1994-09
影响因子：
0.4
通讯作者：
A. Kawauchi
中科院分区：
文献类型：
--
作者：
A. Kawauchi
证明的思想是研究具有无限循环基本群的闭4-rnan流形之间的同调余边何时是h-余边，而h-余边始终是Freedman[2]的乘积余边，因为我们可以用类似于Kervaire的外科论证的方法来构造M和S x*S3fl：m1之间的同调余边。Freedman在文[1]中证明了任意两个闭有向单连通4-流形Mly M是保定向同胚当且仅当H2(M1‘yz)上的交形同构，且KirbySiebenmann不变量k(7W)，ks(7kf’1)(EZ2)相等，此时存在保定向同胚M±M诱导交形同构.通过将Freedman的这种分类与上面的分裂定理相结合，我们得到了具有无限循环基本群的闭定向4-流形的类似刻画：
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 M 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: