Integral cohomology groups of real toric manifolds and small covers

Abstract

For a simplicial complex $K$ with $m$ vertices, there is a canonical ${\mathbb{Z}}_2^m$-space known as a real moment angle complex $\mathbb{R}{\mathcal{Z}}_K$. In this paper, we consider the quotient spaces $Y=\mathbb{R}{\mathcal{Z}}_K/{\mathbb{Z}}_2^k$, where $K$ is a pure shellable complex and ${\mathbb{Z}}_2^k\subset {\mathbb{Z}}_2^m$ is a maximal free action on $\mathbb{R}{\mathcal{Z}}_K$. A typical example of such spaces is a small cover, where a small cover is known as a topological analog of a real toric manifold. We compute the integral cohomology group of $Y$ by using the PL cell decomposition obtained from a shelling of $K$. In addition, we compute the Bockstein spectral sequence of $Y$ explicitly.

22 pages, 5 figure. The statement of the main theorem has been corrected. Some examples are provided in the last Section

References in corpus (2)

• Multiplication structure of the cohomology ring of real toric spaces
• On the cohomology of polyhedral products with space pairs $(D^1,S^0)$

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• The cohomology groups of real toric varieties associated to Weyl chambers of type $C$ and $D$
• The cohomology rings of real toric spaces and smooth real toric varieties
• Geometric representations of finite groups on real toric spaces
• A Cartan formula for the cohomology of polyhedral products and its application to the ring structure
• On shellability for a poset of even subgraphs of a graph
• Non-isotopic monotone Lagrangian submanifolds of ${\mathbb{C}}^n$
• A moment-angle manifold whose cohomology is not torsion free