#### Volume 17, issue 6 (2017)

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On the integral cohomology ring of toric orbifolds and singular toric varieties

### Anthony Bahri, Soumen Sarkar and Jongbaek Song

Algebraic & Geometric Topology 17 (2017) 3779–3810
##### Abstract

We examine the integral cohomology rings of certain families of $2n$–dimensional orbifolds $X$ that are equipped with a well-behaved action of the $n$–dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs $\left(Q,\lambda \right)$, where $Q$ is a simple convex $n$–polytope and $\lambda$ a labeling of its facets, and from $n$–dimensional fans $\Sigma$. In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on $\left(Q,\lambda \right)$ or on $\Sigma$ which ensure that the integral cohomology groups ${H}^{\ast }\left(X\right)$ of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring ${H}^{\ast }\left(X\right)$ as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.

##### Keywords
toric orbifold, quasitoric orbifold, toric variety, lens space, equivariant cohomology, Stanley–Reisner ring, piecewise polynomial
##### Mathematical Subject Classification 2010
Primary: 14M25, 55N91, 57R18
Secondary: 13F55, 52B11