# 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 $\mathrm{mod}$ $2$ 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) $\mathrm{mod}$ $2$ Bockstein homomorphisms on the $\mathrm{mod}$ $2$ 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.