The Second Stiefel-Whitney Class of Small Covers

Abstract

Let π: M ⁿ → P ⁿ be an n-dimensional small cover over P ⁿ and λ: ℱ(P ⁿ) → ℤ₂ ⁿ be its characteristic function. The author uses the symbol c(λ) to denote the cardinal number of the image Im(λ). If c(λ) = n + 1 or n + 2, then a necessary and sufficient condition on the existence of spin structure on M ⁿ is given. As a byproduct, under some special conditions, the author uses the second Stiefel-Whitney class to detect when P ⁿ is n-colorable or (n + 1)-colorable.

Keywords

Small cover / Spin structure / Simple polytope

References

[1] Ayzenberg A. Relation between the Buchstaber invariant and generalized chromatic numbers. Dalarnevost. Mat. Zh., 2011, 11(2): 113-139

[2] Borel A, Hirzebruch F. Characterisic classes and homogeneous spaces I. Amer. J. Math., 1958, 80(2): 458-538

[3] Buchstaber V, Panov T. Torus actions and their applications in topology and combinatorics, 2015, Providence, RI: American Mathematical Society

[4] Davis M, Januszkiewicz T. Convex polytopes, Coxeter orbifolds and torus actions. Duke. Math. J., 1991, 61: 417-451

[5] Erokhovets N. Buchstaber invariant of simple polytopes. Russian. Math. Surveys., 2008, 63(5): 962-964

[6] Joswig M. Projectivities in simplicial complexes and colorings of simple polytopes. Math. Z., 2002, 240(2): 243-259

[7] Lü Z. 2-torus manifolds, cobordism and small covers. Pacific Journal of Mathematics, 2009, 241(2): 285-308

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

[9] Lü Z, Tan Q. Small covers and the equivariant bordism classification of 2-torus manifolds. Int. Math. Res. Not., 2014, 24: 6756-6797

[10] Lü Z, Yu L. Topological types of 3-dimensional small covers. Forum. Math., 2011, 23(2): 245-284

[11] Miller E, Sturmfels B. Combinatorial Commutative Algebra. Graduate Texts in Math, 2005, Berlin: Springer vol. 227

[12] Milnor J, Stasheff J. Characteristic Classes, 1974, Princeton: Princeton University Press

[13] Nakayama H, Nishimura Y. The orientability of small covers and coloring simple polytopes. Osaka. J. Math., 2005, 42(1): 243-256

[14] Nishimura Y. Combinatorial constructions of three-dimensional small covers. Pacific. J. Math., 2012, 256(1): 177-199

[15] Notbohm D. Colorings of simplicial complexes and vector bundles over Davis-Januszkiewicz spaces. Math. Z., 2010, 266(2): 399-405