Surface configuration kernels

TL;DR

This work analyzes the action of the mapping class group on the rational cohomology of configuration spaces on a punctured surface, focusing on the configuration Johnson kernel $J^{cfg}_{g,*}(n)$ and its relationship to the standard Johnson subgroups $J_{g,*}(n)$. Central to the approach is the map $\triangle^n$ from the augmentation-graded group ring $\mathbb{Q}\pi/\mathcal{I}^{n+1}$ to $H^n(Conf_n(\Sigma_{g,*});\mathbb{Q})$, whose kernel encodes $J^{cfg}_{g,*}(n)$ and whose image ${\mathcal{I}}^{cfg}_n$ controls the quotient. The authors develop a granular, representation-theoretic analysis using gr-algebraic structures for $\operatorname{Sp}_{2g}(\mathbb{Z})$-representations, together with a Magnus-type correspondence and chord-diagram techniques to bound and describe the kernel, and they give a geometric construction (the submanifold $\mathcal{E}$) to prove linear independence and a perfect pairing with dual submanifolds. They obtain a virtual cofiltration of $J^{cfg}_{g,*}(n)/J_{g,*}(n)$ with large abelian layers arising from higher Johnson images and symplectic representations, thereby refuting the BMW conjecture in general and proving sharp weight bounds in favorable genus ranges. The results reveal substantial, previously hidden structure in the kernel quotients and connect algebraic weight filtrations with explicit geometric configurations in configuration spaces.

Abstract

Let $Σ_{g,*}$ be a once-punctured oriented surface of genus $g$. We study the action of the mapping class group $Γ_{g,*}$ on the $n^{th}$ rational cohomology of the configuration space $\text{Conf}_n(Σ_{g,*})$ of injections $\{1,\ldots, n\}\hookrightarrow Σ_{g,*}$, and compare the kernel $J_{g,*}^{cfg}(n)$ of this action with the $n^{th}$ Johnson subgroup $J_{g,*}(n)$. We find high-rank abelian subgroups in the quotient $J_{g,*}^{cfg}(n)/J_{g,*}(n)$ arising from the higher Johnson images and from symplectic representation theory. In particular we refute a conjecture due to Bianchi--Miller--Wilson.

Figures (12)

• Figure 1: The closed surface ${\mathcal{X}}$. It is obtained by identifying the pairs of intervals labelled $\alpha_{\pm i}$ together by preserving the diretions of the arrows; the top three sides of the rectangle as well as the $4g+1$ labelled points on the bottom side are all collapsed to the basepoint $p$.
• Figure 2: The dual curves $\beta_{\pm i}$, the intersection points $p_i$, and the paths $\epsilon_{i,i+1}$ in the open surface $U$. (Here only a part of $U$ is depicted.)
• Figure 3: A $5$-torus in $\mathscr{C}\!\mathit{onf}\!_5(U)$. The submanifold $\gamma_1^{(2)}\times \gamma_2^{(5)}\times \gamma_3^{(1)}\times \gamma_4^{(4)}\times \gamma_5^{(3)}$ intersects $\mathscr{C}\!\mathit{onf}\!_5(U_1)$ precisely once transversally in the component $\triangle^{(2,5,1)}(\alpha_{-1})\times \triangle^{(4,3)}(\alpha_2)$.
• Figure 4: The embedded torus-minus-a-disc $\Sigma^{(1,2)}(\beta_i\times \beta_{-i}$).
• Figure 5: An animation of the tube $v_{1,2}(\epsilon)$: particle $1$ is traversing arc $\epsilon$ while particle $2$ is orbiting $1$ at a fixed radius $\varepsilon$.

Theorems & Definitions (73)

• Theorem A
• Remark 1.1
• Theorem B
• Remark 1.2
• Theorem C: Theorem \ref{['thm:assgradedIcalcfg']}
• Theorem D: Theorem \ref{['thm:virtualcofiltration']}
• Theorem 2.1: Theorem 2.10 LooijengaStavrou25
• Proposition 2.2: Corollary 3.9 LooijengaStavrou25
• Definition 2.3
• Example 2.4