Full subcomplexes of Bier spheres

Full subcomplexes of a simplicial complex encode essential structure for understanding the complex itself. For a simplicial complex $K$, possibly with a ghost vertex, the Bier sphere of $K$ is a simplicial sphere obtained as the deleted join of $K$ and its combinatorial Alexander dual. In this paper, we determine the homotopy types of all full subcomplexes of Bier spheres. As applications, we provide a formula for the bigraded Betti numbers of the Bier sphere of $K$ in terms of full subcomplexes of $K$, and we explicitly describe the cohomology of real toric manifolds associated with Bier spheres.