On the low-dimensional homology of SL2(k[t,t−1])

We prove analogues of the fundamental theorem of K-theory for the second and the third homology of SL2 over an infinite field. The statements of the theorems involve Milnor-Witt K-theory and refined scissors congruence groups. We use these results to calculate the low-dimensional homology of SL2 of Laurent polynomials over certain fields.