On the homology of the orthogonal coefficient group in Clifford algebras

The object of this Note is to show that the method used by Dupont and Sah to compute the homology groups H-1(SO(3;R),so(3;R)) and H-2(SO(3;R),so(3;R)) can be reformulated more generally in terms of non-commutative differential forms over Clifford algebras. Applying then this formalism to other Clifford algebras, we are able on the one hand to retrieve the results of Cathelineau for the groups H-1(SL2(C), sl(2)(C) and H-2(SL2(C),sl(2)(C)), and on the other hand to compute H-1(SL2)(R), sl(2)(R)) and H-2(SL2(R),sl(2)(R)), which are isomorphic to Omega(R\Q)(1) and to the null group respectively. (C) 2003 Academie des sciences/Editions scientifiques et medicales Elsevier SAS. All rights reserved.