1,4

A small cover, as defined by Davis and Januszkiewicz, is an n-dimensional closed smooth manifold M with a smooth action of standard real torus (Z_2)^2 action such that the action is locally isomorphic to a standard action of (Z_2)^2 on R^n and the orbit space M/(Z_2)^2 is a simple convex polytope. For instance, RP^n with a natural action of (Z_2)^2 is a small cover over an n-simplex. In general, real toric manifolds, the set of real points of a toric manifold, provide examples of small covers.

Hence we may think of small covers as a topological generalization of real toric manifolds in algebraic geometry. Small covers over hypercubes are known to be real Bott manifolds, which is obtained as iterated RP^1 bundles starting with a point, where each fibration is the projectivization of a Whitney sum of two real line bundles [Masuda and Panov]. Choi found the 1-1 correspondence between the set of real Bott manifolds and the set of acyclic digraphs in a previous work.

Suyoung Choi. The number of small covers over cubes. Algebr. Geom. Topol. (to appear).

Michael W. Davis and Tadeusz Januszkiewicz. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J., 62(2):417-451, 1991.

Mikiya Masuda and Taras E. Panov. Semifree circle actions, Bott towers and quasitoric manifolds. Sbornik Math., 199(8):1201-1223, 2008.

Suyoung Choi, The Number of orientable small covers over cubes, Dec 19, 2008.

Sum_{n>0} a(n)*x^n/(2^(n-1)*x-1)^(n+1) = x/(1-x). [From Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 24 2009]

Sequence in context: A027311 A074702 A152282 this_sequence A015084 A102388 A071125

Adjacent sequences: A153252 A153253 A153254 this_sequence A153256 A153257 A153258

nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 21 2008

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 24 2009