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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A geometric interpretation of the Chern classes
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by R. Sivera Villanueva PDF
Trans. Amer. Math. Soc. 276 (1983), 193-200 Request permission

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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 276 (1983), 193-200
  • MSC: Primary 57R20; Secondary 55P47
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0684502-3
  • MathSciNet review: 684502