Cellular Chain Complex of Small Covers with Integer Coefficients and Its Application
Cellular Chain Complex of Small Covers with Integer Coefficients and Its Application
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DOI：
10.1007/s10114-018-7160-4
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发表时间：
2018-02
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通讯作者：
D. Liu
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作者：
D. Liu
这是一个具有z2特征函数λ的简单多面体。这是摩尔斯函数除以pn。那么对应于这对（Pn,λ）的小的coverMn（λ）具有由h给出的细胞结构。从这种细胞结构中，我们可以得到一个具有整数系数的mn （λ）的细胞链复合体。在本文中，我们首先讨论了这个元胞链复合体的最高维边界态射∂n，并通过自然的方法得到了∂n= 0或2。然后，从众所周知的结果（F,λF）对应的子流形自然是一个维度为k的小覆盖，其中F是pnk的任意面，λF是λ onf的限制，我们得到∂k= 0或±2，当0≤k<n时。最后，利用遗传特征函数的定义，即λ在pn面上的约束，得到了mn （λ）的同调群的计算方法。将我们的结果应用到3-小盖上，我们得到任何3-小盖上的同调群都是无扭转的或者只有2-扭转。
LetPnbe a simplen-polytope with a Z2-characteristic functionλ. Andhis a Morse function overPn. Then the small coverMn(λ) corresponding to the pair (Pn,λ) has a cell structure given byh. From this cell structure we can derive a cellular chain complex ofMn(λ) with integer coefficients. In this paper, firstly, we discuss the highest dimensional boundary morphism∂nof 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 dimensionk, whereFis anyk-face ofPnandλFis the restriction ofλonF, we get that∂k= 0 or ±2 for 0 ≤k<n. Finally, by using the definition of inherited characteristic function which is the restriction ofλon the faces ofPn, we get a way to calculate the homology groups ofMn(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.