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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A geometric interpretation of the Chern classes

Author: R. Sivera Villanueva
Journal: Trans. Amer. Math. Soc. 276 (1983), 193-200
MSC: Primary 57R20; Secondary 55P47
MathSciNet review: 684502
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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 $g,g_k^{\ast }([{N_k}])$.

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Keywords: Stable complex vector bundle, Chern classes, infinite loop space, cobordism of immersions, <IMG WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$k$">-tuple points
Article copyright: © Copyright 1983 American Mathematical Society