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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Function spaces of CW homotopy type are Hilbert manifolds


Authors: Jaka Smrekar and Atsushi Yamashita
Journal: Proc. Amer. Math. Soc. 137 (2009), 751-759
MSC (2000): Primary 54C35; Secondary 55M15, 57N20
Published electronically: August 28, 2008
MathSciNet review: 2448598
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Abstract: Let $ X$ be a countable CW complex and $ Y$ an ANR (for metric spaces) and let $ Y^X$ denote the space of continuous maps from $ X$ to $ Y$ with the compact-open topology. We show that, under mild restrictions, the following are equivalent: (1) $ Y^X$ is an $ \ell^2$-manifold, (2) $ Y^X$ is an ANR, (3) $ Y^X$ has the homotopy type of a CW complex. We also give a few interesting examples and applications.


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Additional Information

Jaka Smrekar
Affiliation: Fakulteta za Matematiko in Fiziko, Jadranska ul. 19, SI-1111 Ljubljana, Slovenia
Email: jaka.smrekar@fmf.uni-lj.si

Atsushi Yamashita
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8914, Japan
Email: yonster@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09584-1
PII: S 0002-9939(08)09584-1
Received by editor(s): February 1, 2008
Published electronically: August 28, 2008
Additional Notes: The first author was supported in part by the ARRS research project No. J1-6128-0101-04.
The second author was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.