Homology of configuration spaces of surfaces modulo an odd prime

2023/07/17 by Andrea Bianchi, Bianchi, Andrea, Andreas Stavrou +1 · 1 citation
Mathematics · #16E30 #20F36 #55R35 #55R80 #55U25 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2307.08664

openalex publication_date 2023/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a compact orientable surface Σ_g,1 of genus g with one boundary component and for an odd prime number p, we study the homology of the unordered configuration spaces C_\bullet(Σ_g,1):=\coprod_n≥0C_n(Σ_g,1) with coefficients in \mathbbF_p. We describe H_*(C_\bullet(Σ_g,1);\mathbbF_p) as a bigraded module over the Pontryagin ring H_*(C_\bullet(D);\mathbbF_p), where D is a disc, and compute in particular the bigraded dimension over \mathbbF_p. We also consider the action of the mapping class group Γ_g,1, and prove that the mod-p Johnson kernel K_g,1(p)⊆Γ_g,1 is the kernel of the action on H_*(C_\bullet(Σ_g,1;\mathbbF_p)).

Cited by

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

Related

• Surface configuration kernels
• Mod p homology of unordered configuration spaces of surfaces
• Non-trivial action of the Johnson filtration on the homology of configuration spaces
• On the mapping class group action on the homology of surface covers
• Cohomology of configuration spaces of surfaces as mapping class group representations