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

 

Fold maps, framed immersions and smooth structures


Author: R. Sadykov
Journal: Trans. Amer. Math. Soc. 364 (2012), 2193-2212
MSC (2010): Primary 57R45; Secondary 55N22
Published electronically: November 10, 2011
MathSciNet review: 2869203
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Abstract: For each integer $ q\ge 0$, there is a cohomology theory $ \mathbf {A}_1$ such that the zero cohomology group $ \mathbf {A}_1^0(N)$ of a manifold $ N$ of dimension $ n$ is a certain group of cobordism classes of proper fold maps of manifolds of dimension $ n+q$ into $ N$. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even $ q$, the splitting theorem implies that the cobordism group of fold maps to a manifold $ N$ is a sum of $ q/2$ cobordism groups of framed immersions to $ N$ and a group related to diffeomorphism groups of manifolds of dimension $ q+1$. Similarly, in the case of odd $ q$, the cobordism group of fold maps splits off $ (q-1)/2$ cobordism groups of framed immersions.

The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds.


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Additional Information

R. Sadykov
Affiliation: Departamento de Matemáticas, CINVESTAV-IPN, A.P. 14-740, C.P. 07000, México, D.F., México
Email: rstsdk@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05485-1
PII: S 0002-9947(2011)05485-1
Received by editor(s): February 16, 2010
Received by editor(s) in revised form: June 21, 2010, and October 8, 2010
Published electronically: November 10, 2011
Additional Notes: The author was supported by the FY2005 Postdoctoral Fellowship for Foreign Researchers of the Japan Society for the Promotion of Science and by a Postdoctoral Fellowship of the Max Planck Institute, Germany
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.