An overview of Euler class theory

The Euler class program was outlined, by M. V. Nori around 1990, as a possible obstruction theory for projective modules of top rank to split o ® a free direct summand. The program has its genesis in Topology and has been the most important development in the subject of projective modules in its recent history. We give an overview of this theory with an emphasis on the following question: Question: Let X = Spec(A) be a smooth a±ne variety of dimension n ¸ 2 over R (the ¯eld of real numbers) and P a projective A¡module of rank n: Under what further restrictions, does vanishing of the top Chern class Cn(P) = 0 =) P ' A©Q? We answer this question completely. We show that, in some cases, addi-tional topological obstruction does exist. This is a joint work with S. M. Bhatwadekar and Mrinal Kanti Das ([BDM], Invent. Math., to appear). The talk is adapted from my talk in Chicago.