Orientable small covers over a product space

Abstract

A small cover is a closed manifold M ⁿ with a locally standard (ℤ₂) ⁿ-action such that its orbit space is a simple convex polytope P ⁿ. Let Δ ⁿ denote an n-simplex and P(m) an m-gon. This paper gives formulas for calculating the number of D-J equivalent classes and equivariant homeomorphism classes of orientable small covers over the product space $\Delta ^{n_1 } \times \Delta ^{n_2 } \times P(m)$, where n ₁ is odd.

Keywords

(ℤ₂) ⁿ-Action / Small cover / Equivariant homeomorphism / Polytope

References

[1] Alperin J. L., Bell R. B.. Groups and Representations, 1995, Berlin: Springer-Verlag

[2] Cai M., Chen X., Lü Z.. Small covers over prisms. Topology Appl., 2007, 154: 2228-2234

[3] Chen Y., Wang Y.. Small covers over a product of simplices. Filomat, 2013, 27: 777-787

[4] Choi S.. The number of orientable small covers over cubes. Proc. Japan Acad., Ser. A, 2010, 86: 97-100

[5] Davis M., Januszkiewicz T.. Convex polytopes. Coxeter orbifolds and torus actions, Duke Math. J., 1991, 62: 417-451

[6] Garrison A., Scott R.. Small covers of the dodecahedron and the 120-cell. Proc. Amer. Math. Soc., 2003, 131: 963-971

[7] Lü Z., Masuda M.. Equivariant classification of 2-torus manifolds. Colloq. Math., 2009, 115: 171-188

[8] Meng Y.. On closed manifolds and moment-angle complex in toric topology, Dissertation, 2012

[9] Nakayama H., Nishimura Y.. The orientability of small covers and coloring simple polytopes. Osaka J. Math., 2005, 42: 243-256

[10] Ziegler G. M.. Lectures on Polytopes, 1994, Berlin: Springer-Verlag