Low-dimensional Homology of SL_2(k[t,1/t])

TL;DR

This paper establishes new results relating the second and third homology groups of SL_2 over fields to Milnor-Witt K-theory and scissors congruence groups, and applies these to Laurent polynomial rings.

Contribution

It provides the first analogues of the fundamental theorem of algebraic K-theory for low-dimensional homology of SL_2 over infinite fields.

Findings

Calculated low-dimensional homology of SL_2 over Laurent polynomial rings.

Connected homology groups to Milnor-Witt K-theory and scissors congruence groups.

Extended algebraic K-theory results to new classes of rings.

Abstract

We prove analogues of the fundamental theorem of algebraic K-theory for the second and third homology of SL_2 over an infinite field k. The statements involve Milnor-Witt K-theory and scissors congruence groups. We use these results to calculate the low-dimensional homology of SL_2 of Laurent polynomials over certain fields.