The third homology of SL_2 of fields with discrete valuation

TL;DR

This paper investigates the third homology group of SL_2 over fields with discrete valuations, linking it to residue fields and scissors congruence groups, and provides explicit calculations for higher-dimensional local fields.

Contribution

It establishes a relationship between the third homology of SL_2 over valued fields and residue fields, introducing refined scissors congruence groups for explicit computations.

Findings

Explicit formulas for third homology of SL_2 in higher-dimensional local fields

Connection between homology groups and scissors congruence groups

New computational methods for homology in valued fields

Abstract

For a field F with discrete valuation and residue field $k$ we relate the third homology of SL_2(F) with half-integral coefficients to the third homology of SL_2(k) and a certain refined scissors congruence group of k. As an application, we obtain explicit calculations of the third homology of SL_2 of certain higher-dimensional local fields in terms of scissors congruence groups, in the sense of Dupont and Sah, of the residue fields of the associated valuations.