二次曲面的交、矩角流形与连通和
Intersections of Quadrics, Moment-angle Manifolds and Connected Sums
摘要：
第二作者研究了两个实齐次共轴二次曲面的交的拓扑，证明了它与单位球面的交在大多数情况下是一个球面乘积的连通和。结合这种方法与最近的一个（由于安东尼巴赫里，马丁Bendersky，弗雷德科恩和第一作者），我们在这里研究k> 2二次曲面的交叉，我们确定了非常一般的家庭，这种流形是同构的连接和领域的产品。这些包括那些矩角流形，其结果是由弗雷德里克·博西奥和劳伦特·梅尔塞曼证明的。作为副产品，对于k = 2的情况，获得了结果的更简单和更整洁的证明。两个新的部分包含了本文第一版中没有的结果：第2部分描述了在切掉相关多面体的一个顶点或一条边之后流形上的拓扑变化，它可以以一种特殊的方式与前面的结果相结合，产生新的无穷流形族，这些流形族是球面积的连通和。在其他情况下，我们得到稍微复杂的流形：有了这个，我们解决了另一个问题的Bosio-Meerseman关于流形相关的截断立方体。在第3节中，我们用它来表明，在一般情况下，矩角流形的上同调积的已知规则必须彻底修改。我们声明修改后的规则，但将其细节留给另一个出版物。第0节回顾了已知的定义和结果，在第2.1节中定义和探讨了一些基本的拓扑结构。在附录中，我们陈述并证明了在第1节和第2节中用到的关于特殊微分流形的一些结果。
英文摘要：
The topology of the intersection of two real homogeneous coaxial quadrics was studied by the second author who showed that its intersection with the unit sphere is in most cases diffeomorphic to a connected sum of sphere products. Combining that approach with a recent one (due to Antony Bahri, Martin Bendersky, Fred Cohen and the first author) we study here the intersections of k>2 quadrics and we identify very general families of such manifolds that are diffeomorphic to connected sums of sphere products. These include those moment-angle manifolds for which the result was conjectured by Frederic Bosio and Laurent Meersseman. As a byproduct, a simpler and neater proof of the result for the case k=2 is obtained. Two new sections contain results not included in the first version of this article: Section 2 describes the topological change on the manifolds after the operations of cutting off a vertex or an edge of the associated polytope, which can be combined in a special way with the previos results to produce new infinite families of manifolds that are connected sums of sphere products. In other cases we get slightly more complicated manifolds: with this we solve another question by Bosio-Meersseman about the manifold associated to the truncated cube. In Section 3 we use this to show that the known rules for the cohomology product of a moment-angle manifold have to be drastically modified in the general situation. We state the modified rule, but leave the details of this for another publication. Section 0 recalls known definitions and results and in section 2.1 some elementary topological constructions are defined and explored. In the Appendix we state and prove some results about specific differentiable manifolds, which are used in sections 1 and 2.
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