Topology of T-Varieties

A $\mathbb{T}$-variety is a variety endowed with an effective action of an algebraic torus $\mathbb{T}$. We use the combinatorial description of $\mathbb{T}$-varieties by means of divisorial fans. In this paper we study the topology of $\mathbb{T}$-varieties in terms of this description. We compute the fundamental group of a $\mathbb{T}$-variety with log-terminal singularities, the cohomology groups of an affine complexity-one $\mathbb{T}$-variety and the cohomology ring and Chow ring of a shellable, complete, $\mathbb{Q}$-factorial, complexity-one $\mathbb{T}$-variety.