Classification Of Orientable Small Covers Over A Product Space

In toric topology, an n-dimensional closed manifold is said to be a small cover if it admits a locally standard (Z2)n-ction such that its orbit space is a simple convex n-polytope. Let △n be a n-simplex and P{m) an m-polygon. By using combinatorial properties of the simple convex polytope △n × △n2 x P(m) where n1 ≥ n2≥ 1, we obtain recursive formulas for calculation of the number of equivariant homeomorphism classes and D-J equivalence classes of the orientable small covers over it. The recursive formulas depend on the values of n1,n2 and m. The discussion consists of four parts.In the first part, we recall some definitions and properties of small covers over a simple convex polytope Pn, including an orientability condition for a small cover. Let O(Pn) be the set of orientable characteristic function and F(Pn) the automorphism group of face poset of Pn. Then, we discuss a GL(n, Z2)-action on O(Pn) and an F(Pn)-action on O(Pn). D-J equivalence class and equivariant homeomorphism class of an orientable small cover over Pn agree with equivalence class of its corresponding orientable characteristic function under GL(n,Z2)-action and F(Pn)-action respectively.The second part is to give recursive formulas for calculation of the number of equiv-ariant homeomorphism classes and D-J equivalence classes of the orientable small covers over △n1 × △n2 x P(m) for n1 odd. The first key point is to find recursive relations in counting the number of equivalence classes of the orientable characteristic functions under the GL(n,Z2)-action. The second key point is to find a system of generators for group F(Pn) so that the number of equivalence classes of the orientable characteristic functions under the F(Pn)-action can be calculated through Burnside Lemma.The third part is to consider the case: n1even and n2 odd. The difference between the second part and the third part is to find new recursive rules in counting the number of equivalence classes of the orientable characteristic functions under the GL(n,Z2)-action.The fourth part is to consider the case: n1 even and n2 even. Compared with the previous parts, a new method is introduced to find recursive rules for calculating the number of D-J equivalence classes of the orientable characteristic functions under the GL(n,Z2)-action.