Torus orbifolds with two fixed points (Q2312906)

Torus orbifolds are \(2n\)-dimensional closed oriented orbifolds with effective actions of \(n\)-dimensional tori with nonempty fixed points. They form a category of a wide range of orbifolds including symplectic toric orbifolds and toric varieties having finite quotient singularities. The authors of this paper consider in particular torus orbifolds with exactly two fixed points. Two dimensional examples of these orbifolds are often called spindles \(S^2(m,n)\) whose underlying topological spaces are homeomorphic to the usual \(2\)-spheres. However, they may have two singular points at the north pole and the south pole, which distinguishes them as orbifolds. The paper begins with a combinatorial model of the orbit space with respect to the torus action, what the authors call a manifold with two vertices (in Section~2). It is combinatorially equivalent to the suspension of a simplex, whose two cone points correspond to the fixed points of the given torus orbifold. Then, they incorporate a manifold with two vertices together with an integral nonsingular square matrix to realize a torus orbifold with two fixed points as a quotient construction. The first main result of this paper (Theorem~4.2) verifies the equivariant homeomorphism type of a torus orbifold of dimension \(2n\) \((n \geq 1)\) with two fixed points. As a corollary, the authors also give a combinatorial condition for a torus orbifold with two fixed points to be homeomorphic to the even-sphere together with the standard torus action on it. The second part of this paper studies the integral equivariant cohomology ring of a torus orbifold with two fixed points. The main source of the computation is the same authors' paper [``The equivariant cohomology of torus orbifolds'', Canad. J. Math. (to appear), \url{doi:10.4153/S0008414X20000760}], where they study the integral GKM-theory for torus orbifolds. Also [loc. cit.] provides another algebraic presentation of the resulting GKM-theory in terms of generators and relations, which is called a weighted face ring. The second main result of this paper applies the result of [loc. cit.] to provide an explicit generators and relations type presentation of the weighted face ring of the combinatorial model of a torus orbifold with two fixed points. For the entire collection see [Zbl 1411.55001].