多面体截面和小覆盖的闭子流形

设π:M~n→P~n是P~n上的小覆盖,S是P~n的任意一个n-1维截面.给出了π~(-1)(S)是n-1维闭子流形(或者两个相互同胚n-1维闭子流形的不交并),以及π~(-1)(S)是n-1维伪流形的充要条件.

Section of Polytope and Closed Submanifold of Small Cover

Let $\pi:M^{n} \rightarrow P^n$ be a small cover of $P^{n}$, $S$ an $(n-1)$-dimensional section of $P^{n}$. The author deals with the relationship between $S$ and $\pi^{-1}(S)$, and obtains a necessary and sufficient condition to guarantee that $\pi^{-1}(S)$ is an $(n-1)$-dimensional closed submanifold (or the disjoint union of two $(n-1)$-dimensional closed submanifolds which are homeomorphic to each other), and a necessary and sufficient condition to guarantee that $\pi^{-1}(S)$ is an $(n-1)$-dimensional pseudomanifold.