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

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, λ_F) is naturally a small cover with dimension k, where F is any k-face of Pⁿ and λ_F 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.