The Homology Groups Of Small Cover Over L???bell Polyhedron L_?3? And Truncated Octahedron

The torus manifolds are an important category in algebraic topology.The study of cohomology groups?rings?on small cover plays an important role in understanding such manifolds.In this paper,for the simple convex polytope:L_?3?and truncated octahedron,we calculate the cohomology group of small cover on them.Let ?:Mⁿ?Pⁿ be the corresponding small cover.First,according to the Morse function on the convex polytope Pⁿ,we can give the cell decomposition of the corresponding small cover Mⁿ over Pⁿ,and the cellular chain complex { ???_i(Mⁿ???,???_i} of Mⁿ.Second,considering the relationship between the boundary homomorphism { ???_i } and the characteristic function ?,the principle of how to determine the boundary homomorphism is given.Finally,the homology groups are computed by { H_i???ker???_i/Im???_i+1},and the corresponding results are given.