Odd-primary torsion in the homology of unordered configurations on the torus

2026/07/26 by Najib Idrissi, Victor Roca i Lucio
#math.AT

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Abstract

Let B_k(Σ₁) denote the unordered configuration space of k points in the torus Σ₁. For every odd prime p, we prove that H_*(B_k(Σ₁);ℤ) has no p-torsion for k≤ 2p-1. At the threshold k=2p, we prove that H_2p-2(B_2p(Σ₁);ℤ) has p-torsion if and only if p≥ 5. For p≥5, the class is the image under puncture filling of the unique Bianchi--Stavrou order-p class on the once-punctured torus; for p=3, Napolitano's calculation shows that this punctured class dies after filling. We also prove that H_2p(B_2p(Σ₁);ℤ) and H_2p+1(B_2p(Σ₁);ℤ) have no p-torsion for every odd p.

Citations

• Representation asymptotics in the homology of pure graph braid groups
• Homology of configuration spaces of surfaces modulo an odd prime
• Quillen homology of spectral Lie algebras with application to mod p homology of labeled configuration spaces
• Computing cohomology of configuration spaces
• Betti numbers of unordered configuration spaces of the torus
• Two mod-p Johnson filtrations
• Characteristics of graph braid groups
• ON THE COHOMOLOGY OF CONFIGURATION SPACES ON SURFACES
• Decomposition of tensor products of modular irreducibles for \SL₂

Related

• Mod p homology of unordered configuration spaces of surfaces
• Torsion primes in loop space homology
• Torsion in the space of commuting elements in a Lie group
• Secondary representation stability and the ordered configuration space of the once-punctured torus
• On odd torsion in even Khovanov homology