2-Sphere Bundles Over Compact Surfaces
2-Sphere Bundles Over Compact Surfaces
复制标题
紧凑表面上的 2 球体束
DOI：
10.2307/2045428
复制
发表时间：
1984
影响因子：
1
通讯作者：
P. Melvin
中科院分区：
文献类型：
--
作者：
P. Melvin
对在紧致曲面上纤维化且纤维化为球面的闭4-流形进行了分类，证明了其纤维化是唯一的（直到复同态）。众所周知，在给定的紧致曲面上至多存在两个可定向的四维流形，它们纤维于2-球面S2。（如果曲面有非空边界，则只有一个;如果曲面是闭合的，则只有两个。如果可定向性条件被放弃，那么情况变得更加复杂。特别是（mod 2）交叉配对不再足以区分出现的流形。人们还必须考虑？t1作用于g2和周围结构。本文的目的是对紧曲面上的所有S2-丛全空间的4-流形（可定向或不可定向）进行分类。我们将在平稳类别中工作。由于Diff（S2）变形收缩到0（3），我们可以假设所有产生的丛都有0（3）作为结构群。沿着的方式，它表明，束结构是唯一的。也就是说，如果任何两个4-流形，如上所述的纤维化，是复同构的，那么它们之间存在一个保持复同构的纤维，它在纤维上正交。我们对S2-丛的兴趣源于对李群作用（特别是S0（3））在4-流形上的作用的研究。这里得到的结果用于等变分类的行动（MP）。
Closed 4-manifolds which fiber over a compact surface with fiber a sphere are classified, and the fibration is shown to be unique (up to diffeomorphism). It is well known that there are at most two orientable 4-manifolds which fiber over a given compact surface with fiber the 2-sphere S2. (There is exactly one if the surface has nonempty boundary, and two if it is closed.) If the orientability condition is dropped, then the situation becomes more involved. In particular the (mod 2) intersection pairing is no longer sufficient to distinguish among the mani- folds that arise. One must also consider the ?Tl-action on g2 and the peripheral structure. The purpose of this note is to classify all 4-manifolds (orientable or not) which are total spaces of S2-bundles over compact surfaces. We shall work in the smooth category. Since Diff(S2) deformation retracts to 0(3), we may assume that all bundles that arise have 0(3) as structure group. Along the way it is shown that the bundle structures are unique. That is, if any two 4-manifolds, fibered as above, are diffeomorphic, then there is a fiber preserving diffeomorphism between them which is orthogonal on fibers. Our interest in S2-bundles arose in the study of Lie group actions (in particular of S0(3)) on 4-manifolds. The results obtained here are used in the equivariant classification of such actions (MP).