Cellular Chain Complex of Small Covers with Integer Coefficients and Its Application

作     者：Deng Pin LIU 

作者机构：Department of Mathematics Guangxi Normal University 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2018年第34卷第11期

页      面：1742-1754页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Project supported by NSFC(Grant No.11401118) the program on the high level innovation team and outstanding scholars in universities of Guangxi province 

主　　题：Small cover homology group orientation CW-complex 

摘      要：Let Pn be a simple n-polytope with a Z2-characteristic function λ. And h is a Morse function over Pn. Then the small cover Mn（λ） corresponding to the pair （Pn, λ） has a cell structure given by h. From this cell structure we can derive a cellular chain complex of Mn（λ） with integer coefficients. In this paper, firstly, we discuss the highest dimensional boundary morphism ？n of this cellular chain complex and get that ？n=0 or 2 by a natural way. And then, from the well-known result that the submanifold corresponding to （F, λF） is naturally a small cover with dimension k, where F is any k-face of Pn and λF is the restriction of λ on F, we get that ？k=0 or ±2 for 0 ≤ k 〈 n. Finally, by using the definition of inherited characteristic function which is the restriction of λ on the faces of Pn, we get a way to calculate the homology groups of Mn（λ）. Applying our result to a 3-small cover we have that the homology groups of any 3-small cover is torsion-free or has only 2-torsion.