The third homology of symplectic groups and algebraic K-theory

TL;DR

This paper advances understanding of the third homology of symplectic groups over certain rings and establishes an isomorphism between Milnor-Witt K-theory and Grothendieck-Witt groups in specific degrees.

Contribution

It improves the homology stability range for the third homology of symplectic groups and proves an isomorphism between Milnor-Witt K-theory and Grothendieck-Witt groups in degrees 2 and 3.

Findings

Enhanced stability range for third homology of symplectic groups.

Proved isomorphism between Milnor-Witt K-theory and Grothendieck-Witt groups in degrees 2 and 3.

Applicable to rings with infinite residue fields of characteristic not 2.

Abstract

We improve the homology stability range for the 3rd integral homology of symplectic groups over commutative local rings with infinite residue field. As an application, we show that for local commutative rings containing an infinite field of characteristic not 2 the symbol map from Milnor-Witt K-theory to higher Grothendieck-Witt groups is an isomorphism in degrees 2 and 3.