On the homology of Lie groups made discrete

Abstract

Here the homology of BG ~ is just the usual Eilenberg-MacLane homology of the uncountably infinite discrete group G 8. These homology groups are of interest in algebraic K-theory (see for example Quillen), in the study of bundles with flat connection (Milnor, 1958), in the theory of foliations (Haefliger, 1973), and also in the study of scissors congruence of polyhedra (Dupont and Sah). They are difficult to compute, and tend to be rather wild. For example if G is non-trivial and connected, then Sah and Wagoner show that HE(BG~;Z) maps onto an uncountable rational vector space. (See also Harris.) The homology and cohomology groups of BG, on the other hand, are much better behaved and better understood. (Borel, 1953.) In w we will see that this Isomorphism Conjecture is true whenever the component of the identity in G is solvable. If it is true for simply-connected simple groups, then it is true for all Lie groups. It is always true for 1-dimensional homology, and is true in a number of interesting special cases for 2-dimensional homology. (See w For higher dimensional computations which tend to support the conjecture, see Karoubi, p. 256, Parry and Sah, as well as Thomason. Another partial result is the following (w If G has only finitely many components, then for any finite coefficient group A the homomorphism H.(BGS; A) --~ H,(BG; A) is split surjective. Thus we obtain a direct sum decomposition