Classification Of Small Covers Over A Product Of A Dual Of A Cyclic Polytope With Simplices

An n-dimensional closed manifold is said to be a small cover if it admits a locally standard (Z2)n-action such that its orbit space is a simple convex n-polytope. Let ?n be an n-simplex and Pmn the dual of the cyclic polytope Cn(m).Because of the properties of the simple convex polytope P63×?N1 × ?n2 in com-binatorics, we obtain formulas for calculations of the number of D-J equivalence classes and equivariant homeomorphism classes of the small covers over it when 1 ? n1 ? 2 and n2 ? 1, and formulas for calculations of the number of D-J equivalence classes and equivariant homeomorphism classes of the orientable small covers over it when n1 = 1.The discussion is divided into four chapters.In the first chapter, we recall definitions and properties of small covers and orientable small covers over Pn. Let A (Pn) and O(Pn) be the set of the characteristic functions and the orientable characteristic functions of Pn respectively. Aut(F(Pn)) denotes the auto-morphism group of the face poset of Pn. It is known that there exist GL(n, Z2)-actions on?(Pn) and O(Pn), and Aut(F(Pn))-actions on A(Pn) and O(Pn). D-J equivalence class and equivariant homeomorphism class of a (orientable) small cover over Pn agree with the orbit space of its corresponding (orientable) characteristic function under GL(n,Z2)-action and Aut(F(Pn))-action respectively.In the second chapter, we determine the number of D-J equivalence classes of all small covers and orientable small covers over P63 × I2. This result could be implemented in the programming language R. Moreover, by finding the generators of the automorphism group Aut(F(P63×I2)) and using Burnside Lemma, we calculate the number of equivariant homeomorphism classes of all small covers and orientable small covers over the product space.In the third chapter, we obtain formulas for calculations of the number of D-J equiv-alence classes of all small covers and orientable small covers over P6 × I × ?n(n ? 2). By considering the Aut(F(P63 × I × ?n))-actions on A(P63 × I × ?n) and O(P63 × I × ?n),we determine the number of equivariant homeomorphism classes of all small covers and orientable small covers over the product space when n?2.The fourth chapter reports the calculation of the number of D-J equivalence classes and equivariant homeomorphism classes of all small covers over P63 × ?2 × ?n(n ? 2).