三维小覆盖的拓扑型
Topological types of 3-dimensional small covers
摘要：
本文研究了一类三维闭流形的（等变）拓扑型，3-维小覆盖），每个小覆盖都有一个局部标准$(\mathbb{Z}_2)^3$作用，使得它的轨道空间是一个简单的凸3-多面体.我们介绍了6个三维小覆盖上的等变运算。这六个操作是有趣的，因为它们的组合性质。然后证明了在这六种操作下，每个3维小覆盖都可以由$\mathbb{R}P^3$和$S^1\times\mathbb{R}P^2$通过一定的$(\mathbb{Z}_2)^3$ -作用得到.作为一个应用，我们分类所有的三维小覆盖$({\Bbb Z}_2)^3$ -等变无向配边。
英文摘要：
In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard $(\mathbb{Z}_2)^3$-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from $\mathbb{R}P^3$ and $S^1\times\mathbb{R}P^2$ with certain $(\mathbb{Z}_2)^3$-actions under these six operations. As an application, we classify all 3-dimensional small covers up to $({\Bbb Z}_2)^3$-equivariant unoriented cobordism.
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