A geometric interpretation of the Chern classes

Abstract

Abstract. Letff M -» BU be a classifying map of the stable complex bundle £ over the weakly complex manifold M. If t is the stable right homotopical inverse of the infinite loop spaces map tj: QBU(\) -» BU, we define/£' = t /{ and we prove that the Chern classes ck(£) are/j'*(A*(ia)), where hk is given by the stable splitting of QBU(\) and tk is the Thorn, class of the bundle y'*' = E1kJf2 yk. Also, we associate to/' an immersion g: N -> M and we prove that ck(í) is the dual of the image of the fundamental class of the £-tuple points manifold of the immersion g, g*([Nk]).1. Introduction. In this paper, we give a geometric interpretation of the Chern classes of a stable complex vector bundle £ over a weakly complex manifold M. In the second section we define characteristic classes ck in H2k(BU; Z) using the weak homotopy equivalence between QBU(\) and CBU(\) for an appropriate coefficient system Q got in [9,4], the stable splitting of CX given in [4] and the stable map t: