The homology groups of small cover on a triangular prism and its number of characteristic functions

Triangular prism is a common geometric shape. From the perspective of algebraic topology, it is a familiar simple convex polyhedron in algebraic topology. In this paper, we mainly calculate that there are only two kinds of characteristic functions on a triangular prism, and the homology groups of triangular prism is obtained by different characteristic functions are different. Firstly, according to the Morse function on the convex polytope Pⁿ, We can give the cell decoposition of the corresponding small cover Mⁿ over Pⁿ, and the cellular chain complex {D_i(Mⁿ(λ)),∂_i} of Mⁿ. Secondly, considering the relationship between the boundary homomorphism {∂_i} and the characteristic function λ, we can give the principle of how to determine the boundary homomorphism is given. Finally, the homology groups are computed by defination {H_i= ker∂_i / Im∂_i+1}, we can give the corresponding results.