[Paper Review] On integral cohomology of certain orbifolds

Anthony Bahri, Dietrich Notbohm+2 more|arXiv (Cornell University)|Nov 6, 2017

Advanced Combinatorial Mathematics17 references3 citations

TL;DR

This paper introduces a q-CW complex framework to detect torsion in the integral cohomology of orbifolds, particularly those with even-dimensional cells. By analyzing attaching maps and group actions on spheres, it proves that if the orders of finite stabilizer groups are coprime to a prime p, then the cohomology has no p-torsion, leading to a complete vanishing of odd-degree integral cohomology under global coprimality conditions.

ABSTRACT

The CW structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this paper we use the concept of $\mathbf{q}$-CW complex structure on an orbifold, to detect torsion in its integral cohomology. The main result can be applied to well known classes of orbifolds or algebraic varieties having orbifold singularities, such as toric orbifolds, simplicial toric varieties, torus orbifolds and weighted Grassmannians.

Motivation & Objective

• To address the difficulty of computing integral cohomology rings for orbifolds, especially when torsion is present.
• To develop a computational framework for orbifolds with complicated CW structures by introducing q-CW complexes.
• To identify verifiable conditions under which the integral cohomology of an orbifold is concentrated in even degrees.
• To generalize prior results on toric orbifolds and weighted Grassmannians by providing a cohomological criterion based on group order coprimality.
• To establish a link between the combinatorics of retraction sequences in polytopes and the absence of torsion in cohomology.

Proposed method

• Introduces q-CW complexes as a generalization of CW complexes, where cells are quotients of discs by finite group actions.
• Defines a building sequence of q-cells attached via maps from quotients of spheres, with special attention to the basepoint 0_i in each q-cell.
• Uses the homotopy type of q-CW complexes to relate them to simplicial complexes via Proposition 2.4.
• Applies the long exact sequence of homology to analyze the effect of attaching q-cells, particularly focusing on the induced maps from the attaching sphere S^{k-1}/G.
• Analyzes the torsion in H_*(S^{k-1}/G; Z) based solely on |G|, showing that p-torsion vanishes if gcd(p, |G|) = 1.
• Applies these results inductively through the building sequence, using exact sequences to propagate the absence of p-torsion through successive skeleta.

Experimental results

Research questions

• RQ1 Under what conditions does the integral cohomology of a q-CW complex have no p-torsion for a given prime p?
• RQ2 Can the absence of odd-degree cohomology in orbifolds be guaranteed by a condition on the orders of finite stabilizer groups?
• RQ3 How does the coprimality of |G_i| with a prime p affect the torsion structure in the cohomology of a q-CW complex?
• RQ4 Can the results on torsion-freeness be applied to known classes of orbifolds such as toric orbifolds and weighted Grassmannians?
• RQ5 What is the relationship between the combinatorics of retraction sequences in polytopes and the cohomological properties of associated orbifolds?

Key findings

• If a q-CW complex has no odd-dimensional q-cells and gcd(p, |G_i|) = 1 for all attaching maps, then H_*(X; Z) has no p-torsion and H_odd(X; Z_p) is trivial.
• When the coprimality condition holds for all primes p, the integral cohomology of the q-CW complex is torsion-free and concentrated in even degrees.
• The main result (Theorem 1.2) provides a sufficient condition for the absence of torsion in the integral cohomology of orbifolds built via q-CW structures.
• For toric orbifolds, the condition gcd(p, |G_i|) = 1 can be checked explicitly from the R-characteristic data (Q, λ), enabling cohomological computations.
• The paper constructs an example showing that the gcd condition in Theorem 4.6 is strictly weaker than the hypothesis in [BSS17, Theorem 1.1], thus generalizing prior results.
• In the case where the attaching map has degree coprime to p and the homology of the skeleton has no p-torsion, the induced map (φ_i)_* preserves torsion-freeness, ensuring no new p-torsion is introduced.