Homological stability for automorphisms of symmetric bilinear forms

2025/12/15 by Vikram Nadig, Nadig, Vikram
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2512.13469

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_{⟨ g,g ⟩}(\mathbb Z) in low degrees.

Citations

• Stable moduli spaces of hermitian forms
• Hermitian K-theory for stable \∞-categories III: Grothendieck-Witt\n groups of rings
• Homological Stability of automorphism groups of quadratic modules and manifolds
• The stable moduli space of Riemann surfaces: Mumford's conjecture

Related

• General symmetric ideals
• Stable homology isomorphisms for the partition and Jones annular algebras
• Homological Stability of automorphism groups of quadratic modules and manifolds
• Stable cohomology of Aut(F_n) with bivariant twisted coefficients
• On the Homological Stability of Orthogonal and Spin Groups