[Paper Review] Homological stability for subgroups of surface braid groups

TriThang Tran|arXiv (Cornell University)|Oct 3, 2014

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper establishes homological stability for subgroups of surface braid groups, specifically the $π_1$ of configuration spaces of $n$ disjoint subsets of $\xi$ points in a surface. Using a new simplicial complex called the 'fern complex' and techniques from homological stability theory, the author proves integral stability for open surfaces and rational stability for both open and closed surfaces, with stability range $2k \leq n$. The results extend classical homological stability to twisted coefficients arising from partitioned point configurations.

ABSTRACT

In this paper we prove homological stability for certain subgroups of surface braid groups. Alternatively, this is equivalent to proving homological stability for configurations of subsets of exactly $ξ$ points in a surface as we increase the number of subsets. For open surfaces, we prove the result integrally using a variation of the arc complex which we dub the "fern complex". We use a technique of Randal-Williams to extend the result rationally for closed surfaces.

Motivation & Objective

• To establish homological stability for subgroups of surface braid groups defined by configurations of $n$ disjoint subsets of exactly $\xi$ points in a surface.
• To extend classical homological stability results to configuration spaces where points are partitioned into labeled subsets of fixed size $\xi$, rather than unstructured point configurations.
• To develop and apply a new simplicial complex—the 'fern complex'—to prove high connectivity, enabling the use of homological stability machinery.
• To prove rational homological stability for closed surfaces using a transfer map technique inspired by Randal-Williams.
• To fill a gap in the literature on homological stability for disconnected manifolds (0-manifolds) in surfaces, where prior dimension conditions do not apply.

Proposed method

• Introduces the 'fern complex'—a variation of the arc complex—on which the $\xi$-configuration braid groups act, and proves its high connectivity via deformation retraction to a contractible space.
• Uses the Hatcher-Wahl axiomatic framework for homological stability to prove integral homological stability for open surfaces, relying on the high connectivity of the fern complex.
• For closed surfaces, applies a rational transfer map construction by adapting techniques from Randal-Williams, using fibrations over configuration spaces of points to define a transfer on homology.
• Employs spectral sequences and Serre spectral sequences to analyze the induced maps on homology, particularly focusing on the $E^1$ and $E^2$ terms to verify isomorphisms in the stability range.
• Uses the Dold-Thom theorem to relate symmetric products of geometric realizations of semi-simplicial spaces to homology, enabling transfer map constructions.
• Establishes weak equivalences between geometric realizations of semi-simplicial spaces and the configuration spaces $\mathrm{Conf}_n^\xi(S)$, allowing homotopical comparison.

Experimental results

Research questions

• RQ1 Does the fundamental group of the space of $n$ disjoint $\xi$-point subsets in a surface satisfy homological stability as $n$ increases?
• RQ2 Can the homological stability of configuration spaces of $n\xi$ points in a surface be extended to the case where the points are partitioned into $n$ labeled subsets of size $\xi$?
• RQ3 Is there a new simplicial complex structure that can replace the arc complex to prove high connectivity for $\xi$-configuration spaces?
• RQ4 Can the rational homological stability result for closed surfaces be established using a transfer map, even when integral stability fails?
• RQ5 Does the theory of homological stability extend to configuration spaces of disconnected manifolds (e.g., 0-manifolds) in surfaces, where prior dimension conditions do not apply?

Key findings

• For open surfaces, the inclusion $\mathrm{Br}_n^\xi(S) \to \mathrm{Br}_{n+1}^\xi(S)$ induces an isomorphism on homology in degrees $k$ such that $2k \leq n$, with integral coefficients.
• For closed surfaces, the rational homology groups $H_k(\mathrm{Br}_n^\xi(S); \mathbb{Q})$ are isomorphic to $H_k(\mathrm{Br}_{n+1}^\xi(S); \mathbb{Q})$ for $2k \leq n$, via a transfer map.
• The 'fern complex'—a new simplicial complex built from arcs avoiding $\xi$-point subsets—is shown to be contractible, implying high connectivity and enabling the stability proof.
• The transfer map $t_n: H_*(\mathrm{Br}_n^\xi(S); \mathbb{Q}) \to H_*(\mathrm{Br}_{n-1}^\xi(S); \mathbb{Q})$ is explicitly constructed and shown to be an isomorphism in the stability range.
• The spectral sequence analysis of the geometric realization of semi-simplicial spaces confirms that the induced map on homology is an isomorphism in degrees $k \leq (n-1)/2$ for rational coefficients.
• The proof establishes that the configuration space $\mathrm{Conf}_n^\xi(S)$ is a $K(\pi,1)$, so its singular homology is isomorphic to the group homology of the $\xi$-braid group $\mathrm{Br}_n^\xi(S)$.