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Homological stability for automorphisms of symmetric bilinear forms

Vikram Nadig

Abstract

We establish homological stability for automorphisms of symmetric bilinear forms over a class of principal ideal domains that includes all fields, the integers, the Gaussian integers, and the Eisenstein integers. In conjunction with Grothendieck-Witt theoretic calculations, this determines a large part of the stable cohomology of the odd orthogonal groups $O_{\langle g,g \rangle}(\mathbb Z)$ in low degrees.

Homological stability for automorphisms of symmetric bilinear forms

Abstract

We establish homological stability for automorphisms of symmetric bilinear forms over a class of principal ideal domains that includes all fields, the integers, the Gaussian integers, and the Eisenstein integers. In conjunction with Grothendieck-Witt theoretic calculations, this determines a large part of the stable cohomology of the odd orthogonal groups in low degrees.

Paper Structure

This paper contains 7 sections, 59 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Combinatorial ingredients
  4. The homological stability machine
  5. Symmetric bilinear forms and associated posets
  6. Connectivity of the space of destabilisations
  7. Applications

Key Result

Theorem 1.2

Let $R$ be a ring satisfying Assumption onlyassint. The following statements are equivalent: In this case, a metabolic form is cofinal if and only if its parity equals $R/(2)$.

Theorems & Definitions (125)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 115 more