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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Local $ H$-maps of $ B{\rm U}$ and applications to smoothing theory


Author: Timothy Lance
Journal: Trans. Amer. Math. Soc. 309 (1988), 391-424
MSC: Primary 55P47; Secondary 55N15, 57R10
MathSciNet review: 957078
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0957078-5
PII: S 0002-9947(1988)0957078-5
Article copyright: © Copyright 1988 American Mathematical Society