Torus Orbifolds with Two Fixed Points

Abstract

The main objects of this paper are torus orbifolds that have exactly two fixed points. We study the equivariant topological type of these orbifolds and consider when we can use the results of [8] to compute its integral equivariant cohomology, in terms of generators and relations, coming from the corresponding orbifold torus graph.

Equivariant cohomology of torus orbifolds

“…Hence, we may think of the S 1 -action on S 2 (m, n) being induced from the standard T 2 /S 1 ⟨m, n⟩ ≅ S 1 -action on S 2 , whose fixed point set consists of two isolated points that are connected by a connected component of one-dimensional orbits. We refer to [GZ01, Example 1.2.1] and [DKS19].…”

“…Remark 2.6 For more on the spindles S 2 (m, n), including describing effective actions when (m, n) are not a pair of coprime integers, see [DKS19].…”

“…Now, the commutativity of (3.8) together with the injectivity of ι 2 establishes h ∈ ker ν. ∎ Remark 3.5 For a calculation of the weighted face ring Z[ , α], when has exactly two vertices, see [DKS19].…”

“…Hence, the resulting torus orbifold X(Q, λ) is exactly the suspension of an orbifold lens space. We refer to [DKS19] for the discussion on these spaces. In particular, if the determinant of Λ is ±1, i.e.…”