On the topology of a small cover associated to a shellable complex
On the topology of a small cover associated to a shellable complex
Small covers are real analogues to quasi-toric manifolds: the fixed points under complex conjugation (i.e., the real toric manifold) in a projective toric manifold is a small cover; Buchstaber and Ray showed that every unoriented cobordism class contains a small cover as its representative. A shelling of a pure simplicial complex K is a special ordering of its facets. If K is a piecewise linear sphere (with a mod 2 characteristic function), such a shelling gives a handle decomposition of the associated small cover M, which is a piecewise linear manifold. With this decomposition, we analyze the cohomology of M with integer coefficients, using (higher) mod 2 Bockstein homomorphisms on the mod 2 cohomology ring of M. As a corollary, we get a necessary and sufficient condition that when M has only torsion-free or 2-torsion elements in cohomology groups. This is a joint work with Suyoung Choi.