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

Let Pn be a simple n-polytope with a Z2-characteristic function .. And h is a Morse function over Pn. Then the small cover Mn(.) corresponding to the pair (Pn, .) has a cell structure given by h. From this cell structure we can derive a cellular chain complex of Mn(.) 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, .F) is naturally a small cover with dimension k, where F is any k-face of Pn and .F is the restriction of . 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 . on the faces of Pn, we get a way to calculate the homology groups of Mn(.). 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.