棱柱上小覆盖的上同调刚性与同胚型数
Cohomological rigidity and the number of homeomorphism types for small covers over prisms
摘要：
本文在M. W. Hirsch（1976）\cite [Chapter 8，§2 Gluing Manifolds Together]{h}，我们给出了构造简单凸多胞体上两个小覆盖之间的同胚的一种方法。因此，我们用$m\geq 3$对三维棱镜$P^3(m)$上的所有小覆盖进行分类，直到同胚。本文引入了两个色棱镜不变量和两个小覆盖的具有${\Bbb Z}_2$ -系数的普通上同调环不变量。在大多数情况下，这些不变量可以构成棱柱上所有小覆盖的同胚类型的完整不变量系统。然后，我们证明了上同调刚性对棱柱$P^3(m)$上的所有小覆盖都成立（即，具有${\Bbb Z}_2$的上同调环-$P^3(m)$上的所有小覆盖的系数确定它们的同胚类型）。此外，我们还计算了$P^3(m)$上所有小覆盖的同胚类型数。
英文摘要：
In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) \cite[Chapter 8, §2 Gluing Manifolds Together]{h}, we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimensional prism $P^3(m)$ with $m\geq 3$. We introduce two invariants from colored prisms and other two invariants from ordinary cohomology rings with ${\Bbb Z}_2$-coefficients of small covers. These invariants can form a complete invariant system of homeomorphism types of all small covers over a prism in most cases. Then we show that the cohomological rigidity holds for all small covers over a prism $P^3(m)$ (i.e., cohomology rings with ${\Bbb Z}_2$-coefficients of all small covers over a $P^3(m)$ determine their homeomorphism types). In addition, we also calculate the number of homeomorphism types of all small covers over $P^3(m)$.
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