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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Local $H$-maps of $B\textrm {U}$ and applications to smoothing theory
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by Timothy Lance PDF
Trans. Amer. Math. Soc. 309 (1988), 391-424 Request permission

Abstract:

When localized at an odd prime $p$, the classifying space $PL/O$ for smoothing theory splits as an infinite loop space into the product $C \times N$ where $C = {\text {Cokernel}} (J)$ and $N$ is the fiber of a $p$-local $H$-map $BU \to BU$. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their $k$-invariants to properties of the defining self-maps of $BU$. If $Y$ is a smooth manifold, the set of homotopy classes $[Y, N]$ is a certain subgroup of resmoothings of $Y$, and the $k$-invariants of $N$ generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of $Y$ and frequently vanish.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 309 (1988), 391-424
  • MSC: Primary 55P47; Secondary 55N15, 57R10
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0957078-5
  • MathSciNet review: 957078