[Paper Review] Homology stability for symplectic groups

Behrooz Mirzaii, Wilberd van der Kallen|arXiv (Cornell University)|Oct 16, 2001

Homotopy and Cohomology in Algebraic Topology23 references6 citations

TL;DR

This paper establishes homology stability for symplectic groups over commutative rings with finite stable rank by proving the high connectivity of the poset of isotropic unimodular sequences, confirming a conjecture by Charney. Using a novel poset nerve theorem and inductive connectivity arguments, the authors show that the poset of such sequences is highly connected, leading to the homology stability result for Sp(2n, R).

ABSTRACT

In this paper the homology stability for symplectic groups over a ring with finite stable rank is established. First we develop a `nerve theorem' on the homotopy type of a poset in terms of a cover by subposets, where the cover is itself indexed by a poset. We use the nerve theorem to show that a poset of sequences of isotropic vectors is highly connected, as conjectured by Charney in the eighties.

Motivation & Objective

• To establish homology stability for symplectic groups over rings with finite stable rank.
• To confirm Charney's conjecture (1980s) that the poset of isotropic unimodular sequences is highly connected.
• To extend Quillen's connectivity theorems to a quantitative poset nerve theorem for homotopy type computation.
• To provide a framework applicable to rings not requiring an infinite field, such as finitely generated Z-algebras.
• To generalize results from orthogonal and symplectic groups over Dedekind domains to more general rings with finite stable rank.

Proposed method

• Develop a new 'nerve theorem' for posets, replacing topological spaces and covers with posets and their order structures.
• Use the poset of sequences satisfying the chain condition as a substitute for simplicial complexes under group actions.
• Apply an inductive argument based on the higher connectivity of unimodular sequence posets (due to van der Kallen).
• Utilize the Hurewicz theorem to relate homological triviality to connectivity of the geometric realization.
• Construct cones and use Mayer-Vietoris and Van Kampen theorems to analyze connectivity of subposets.
• Leverage the fact that every unimodular vector in R^{2n} is isotropic under the standard symplectic form to simplify connectivity arguments.

Experimental results

Research questions

• RQ1 Is the poset of isotropic unimodular sequences over a ring of finite stable rank highly connected?
• RQ2 Can homology stability for symplectic groups over rings with finite stable rank be established via poset connectivity?
• RQ3 Does a quantitative poset nerve theorem enable homotopy type analysis of posets under group actions?
• RQ4 Can the connectivity of the isotropic unimodular sequence poset be proven without assuming the base ring is an infinite field?
• RQ5 To what extent does the structure of the symplectic form (vs. quadratic forms) allow for stronger connectivity results in the symplectic case?

Key findings

• The poset of isotropic unimodular sequences in R^{2n} is at least ⌊(n − sr(R) − 3)/2⌋-connected, where sr(R) is the stable rank of R.
• The poset of unimodular sequences in R^{2n} is known to be highly connected, and this property is used inductively to prove connectivity of the isotropic subposet.
• The authors prove that the poset X of isotropic unimodular sequences is l-connected, where l = min{l−1, l−|v|+1}, under the inductive assumption on lower-dimensional subposets.
• The Mayer-Vietoris sequence and Van Kampen theorem are used to show that the inclusion of subposets induces isomorphisms on homology and trivial fundamental groups, respectively.
• The key result is that the poset of isotropic unimodular sequences is highly connected, confirming Charney’s conjecture for the standard symplectic form.
• As a consequence, the homology stability theorem for symplectic groups holds: H_i(Sp(2n,R), L) → H_i(Sp(2n+2,R), L) is bijective for n ≥ 2i + sr(R) + 3 and surjective for n ≥ 2i + sr(R) + 2.