Cohomology of toric diagrams

Abstract: In this note, integral cohomology of homotopy colimits for toric diagrams and their classifying spaces over regular CW posets are described in terms of sheaf cohomology. Split $T$-CW-complexes with CW orbit poset $C$ have such decomposition (up to a homeomorphism) in terms of a $T$-diagram $D$ over $C$. Equivariant formality for $hocolim\ D$ is equivalent to $H^{odd}(hocolim\ D)=0$ (over $\mathbb{Q}$, or over $\mathbb{Z}$ for connected stabilizers) provided that $C$ is an oriented homology manifold. The integral singular cohomology groups and bigraded Betti numbers are computed in this setting. Similar descriptions are provided for skeleta of toric manifolds and compact nonsingular toric varieties. The cohomological orbit spectral sequence collapse over $\mathbb{Z}$ at page $2$ is proved for any compact toric variety.