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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 ^{\prime} = \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^{\prime}$ 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, $ k$-tuple points
Article copyright: © Copyright 1983 American Mathematical Society