The third homology of SL₂ of fields with discrete valuation

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.