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Homological stability for symplectic groups via algebraic arc complexes

Ismael Sierra, Nathalie Wahl

TL;DR

The paper develops an algebraic framework for homological stability of symplectic groups by replacing the classical direct-sum stabilization with a rank-one stabilization by the object $X=(R,0,\mathrm{id})$ in the braided monoidal category of formed spaces with boundary $\mathcal{F}_\partial$. Central to the approach is the stabilization machine of RWWkrannich, which reduces stability to the high connectivity of a destabilization complex, here realized algebraically as a disordered arc complex $\mathcal{D}(M,\lambda,\partial)$ and its relation to a destabilization complex $W_n(A,X)$. The authors introduce and analyze arc and hyperbolic genera, derive robust cancellation properties under finite $usr(R)$, and show that cutting arcs behaves predictably with respect to genus, enabling precise stability ranges with slope $2/3$ for rings with finite unitary stable rank. They also extend stability to finite-degree and abelian coefficient systems. The results connect algebraic models to geometric intuition from surfaces, and provide explicit stabilization ranges and structural insights for symplectic groups over a broad class of rings, with optimal or near-optimal ranges known in special cases like finite fields or local rings. The work significantly broadens the scope of homological stability in algebraic settings by embedding the problem into an $E_2$-module/$E_1$-algebra action framework and exploiting deep arc-complex connectivity properties.

Abstract

We use algebraic arc complexes to prove a homological stability result for symplectic groups with slope 2/3 for rings with finite unitary stable rank. Symplectic groups are here interpreted as the automorphism groups of formed spaces with boundary, which are algebraic analogues of surfaces with boundary, that we also study in the present paper. Our stabilization map is a rank one stabilization in the category of formed spaces with boundary, going through both odd and even symplectic groups.

Homological stability for symplectic groups via algebraic arc complexes

TL;DR

The paper develops an algebraic framework for homological stability of symplectic groups by replacing the classical direct-sum stabilization with a rank-one stabilization by the object in the braided monoidal category of formed spaces with boundary . Central to the approach is the stabilization machine of RWWkrannich, which reduces stability to the high connectivity of a destabilization complex, here realized algebraically as a disordered arc complex and its relation to a destabilization complex . The authors introduce and analyze arc and hyperbolic genera, derive robust cancellation properties under finite , and show that cutting arcs behaves predictably with respect to genus, enabling precise stability ranges with slope for rings with finite unitary stable rank. They also extend stability to finite-degree and abelian coefficient systems. The results connect algebraic models to geometric intuition from surfaces, and provide explicit stabilization ranges and structural insights for symplectic groups over a broad class of rings, with optimal or near-optimal ranges known in special cases like finite fields or local rings. The work significantly broadens the scope of homological stability in algebraic settings by embedding the problem into an -module/-algebra action framework and exploiting deep arc-complex connectivity properties.

Abstract

We use algebraic arc complexes to prove a homological stability result for symplectic groups with slope 2/3 for rings with finite unitary stable rank. Symplectic groups are here interpreted as the automorphism groups of formed spaces with boundary, which are algebraic analogues of surfaces with boundary, that we also study in the present paper. Our stabilization map is a rank one stabilization in the category of formed spaces with boundary, going through both odd and even symplectic groups.

Paper Structure

This paper contains 20 sections, 35 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formed spaces with boundary
  3. Formed spaces with boundary from bidecorated surfaces
  4. Braided monoidal structure
  5. The hyperbolic genus and the arc genus I
  6. Properties of the base ring
  7. The hyperbolic genus and the arc genus II
  8. Arc complexes
  9. Simplicial complexes, semi-simplicial sets and posets
  10. Complexes of non-separating arcs
  11. Cutting arcs
  12. Disordered arc complexes
  13. Cancellation properties and destabilization complex
  14. Cancellation
  15. The destabilization complex
  16. ...and 5 more sections

Key Result

Theorem 1

Let $R$ be a commutative ring with $usr(R)<\infty$ and set $c=0$ if $R$ is a PID, and $c=2 usr(R)+2$ otherwise. Then the map is an epimorphism for $i\le \frac{n-c}{3}$, and a monomorphism for $i\le \frac{n-c-3}{3}$ ($n$ odd) and for all $i$ ($n$ is even). In particular, restricting to even symplectic groups we get that is an epimorphism for $i \le \frac{2g-c}{3}$ and an isomorphism for $i\le \fr

Figures (2)

  • Figure 1: Two bidecorated surfaces $(S,I_0,I_1)$ and their associated surface $S^+=S\cup H$
  • Figure 2: Transverse representatives of the extended arcs $\rho\cup C$ and $\rho'\cup C$ crossing in $H$

Theorems & Definitions (96)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4: Curved forms
  • Example 2.5
  • Proposition 2.6
  • proof : Proof of Proposition \ref{['prop:Hdmonoidal']}
  • Remark 2.7: Choices of bases for $X^{\# n}$ and their geometric description
  • ...and 86 more