The second homology of SL2SL2 of S-integers

We calculate the structure of the finitely generated groups H2(SL2(Z[1/m]),Z)H2(SL2(Z[1/m]),Z) when m   is a multiple of 6. Furthermore, we show how to construct homology classes, represented by cycles in the bar resolution, which generate these groups and have prescribed orders. When n≥2n≥2 and m is the product of the first n   primes, we combine our results with those of Jun Morita to show that the projection St(2,Z[1/m])→SL2(Z[1/m])St(2,Z[1/m])→SL2(Z[1/m]) is the universal central extension. Our methods have wider applicability: The main result on the structure of the second homology of certain rings is valid for rings of S-integers with sufficiently many units. For a wide class of rings A  , we construct explicit homology classes in H2(SL2(A),Z)H2(SL2(A),Z), functorially dependent on a pair of units, which correspond to symbols in K2(2,A)K2(2,A).