The Homology Groups Of Small Cover Over L?bell Polyhedron L_?5? And The 4-Dimensional Cube

The L?bell polyhedron is an important type of simple convex polytope,which can be given colouring to obtain small cover manifolds.In this paper,we calculate the integer coefficients homology groups of the small cover manifold over Lobell polyhedron L(s).Because of the simplest of cubes,especially as we know the rigidity problem of the small cover over 4-dimensional cube has been solved,so we also consider about the integer coefficients homology groups of the small cover over 4-dimensional cube,and have a detailed calculation.At last,according to the fact that any 3-dimensional simple convex polytope can be obtained from ?3 by cutting point,a edge or two edgs with angle.We can calculate the in-teger coefficient homology groups over the triangular prism and quadrant prisrm,pentaprism prism and hexagonal prism,and get the results of them.