Cohomological rigidity and the number of homeomorphism types for small covers over prisms

In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) [8, Chapter 8, §2 Gluing Manifolds Together], we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimensional prism P3(m) with m⩾3. We introduce two invariants from colored prisms and other two invariants from ordinary cohomology rings with Z2-coefficients of small covers. These invariants can form a complete invariant system of homeomorphism types of all small covers over a prism in most cases. Then we show that the cohomological rigidity holds for all small covers over a prism P3(m) (i.e., cohomology rings with Z2-coefficients of all small covers over a P3(m) determine their homeomorphism types). In addition, we also calculate the number of homeomorphism types of all small covers over P3(m).