A geometric interpretation of the Chern classes

Abstract:

Let ${f_\xi }: M \to BU$ be a classifying map of the stable complex bundle $\xi$ over the weakly complex manifold $M$. If $\tau$ is the stable right homotopical inverse of the infinite loop spaces map $\eta :QBU(1) \to BU$, we define $f_\xi ’ = \tau \cdot {f_\xi }$ and we prove that the Chern classes ${c_k}(\xi )$ are $f_\xi ^{\prime \ast }(h_k^{\ast }(t_k))$, where ${h_k}$ is given by the stable splitting of $QBU(1)$ and ${t_k}$ is the Thom class of the bundle ${\gamma ^{(k)}} = E{\Sigma _k}{X_{{\Sigma _k}}}{\gamma ^k}$. Also, we associate to $f’$ an immersion $g:N \to M$ and we prove that ${c_k}(\xi )$ is the dual of the image of the fundamental class of the $k$-tuple points manifold of the immersion $g,g_k^{\ast }([{N_k}])$.