A Geometric Interpretation of the Chern Classes

Abstract

Let ft: M -* BU be a classifying map of the stable complex bundle t over the weakly complex manifold M. If T is the stable right homotopical inverse of the infinite loop spaces map -q: QBU(1) - BU, we definef = X ft and we prove that the Chem classes Ck(M) are fi*(h*(tk)), where hk is given by the stable splitting of QBU(I) and tk is the Thom,class of the bundle y(k) = E2k Xkyk. Also, we associate to f' an immersion g: N - M and we prove that Ck(A) is the dual of the image of the fundamental class of the k-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 t over a weakly complex manifold M. In the second section we define characteristic classes Ck in H2k(BU; Z) using the weak