Torsion in the cohomology of torus orbifolds

Abstract

We study torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X with actions of the n-dimensional compact torus. Compact simplicial toric varieties are in our family. For a prime number p, we find a necessary condition for the integral cohomology of X to have no p-torsion. Then we prove that the necessary condition is sufficient in some cases. We also give an example of X which shows that the necessary condition is not sufficient in general.

Infinite families of equivariantly formal toric orbifolds

“…Next two examples are applications of Theorem 3.7 to toric orbifolds over a polygon and a simplex, respectively. Though these results are well known from the literature, for example [Fis92,Jor98,KMZ17] and [Kaw73], the same conclusions follow from the results above.…”

The Orientability of Real Toric Manifolds

“…Finally, we remark that this paper has been partially motivated by the paper [4] and the technique of Yeroshkin in [8] that deletes a small neighborhood of the singular set in X(P, λ) in order to obtain a smooth part and investigates the relation of the cohomology groups between X(P, λ) and the smooth part. To be a little more precise, the paper [4] studies the torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X(P, λ) with actions of the n-dimensional compact torus.…”

“…To be a little more precise, the paper [4] studies the torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X(P, λ) with actions of the n-dimensional compact torus. However, it should be also remarked that at the moment there seems to be some problem in [4 The aim of this section is to give a proof of Theorem 1.1. To do so, as before let P be a nice manifold with corners of dimension n ≥ 2 with m facets, and let X R = (P, λ R ) be a real toric manifold for a characteristic function λ R with the quotient map π :…”

“…second) equality. Note that we need the orientabilty of X R \(X R ) • in order to apply the Poincaré-Lefschetz duality as in the above equation (4). Hence by (3) and (4) we have…”

On Integral Cohomology of Certain Orbifolds

“…A torus orbifold X P (λ) from a pair (P, λ), where P is a nice manifold with corners and λ is an R-characteristic function on P , can be defined similarly to the construction of a toric orbifold from an R-characteristic pair, see for example [KMZ,MMP07,MP06] and (4.3) in Subsection 4.1. Moreover, all the notation of the local structures, such as g Q (v) and g E (v) of a toric orbifold in Subsection 4.1 naturally extends to a torus orbifold X P (λ).…”