Function spaces of CW homotopy type are Hilbert manifolds

Smrekar, Jaka; Yamashita, Atsushi

doi:10.1090/S0002-9939-08-09584-1

Function spaces of CW homotopy type are Hilbert manifolds
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by Jaka Smrekar and Atsushi Yamashita

Proc. Amer. Math. Soc. 137 (2009), 751-759

DOI: https://doi.org/10.1090/S0002-9939-08-09584-1

Published electronically: August 28, 2008

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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.

References

Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168
Robert Cauty, Une caractérisation des rétractes absolus de voisinage, Fund. Math. 144 (1994), no. 1, 11–22 (French, with English summary). MR 1271475, DOI 10.4064/fm-144-1-11-22
Tammo tom Dieck, Partitions of unity in homotopy theory, Compositio Math. 23 (1971), 159–167. MR 293625
T. Dobrowolski and H. Toruńczyk, Separable complete ANRs admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), no. 3, 229–235. MR 623731, DOI 10.1016/0166-8641(81)90001-8
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
Sibe Mardešić and Jack Segal, Shape theory, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982. The inverse system approach. MR 676973
C. R. F. Maunder, Algebraic topology, Dover Publications, Inc., Mineola, NY, 1996. Reprint of the 1980 edition. MR 1402473
E. Michael, Uniform ARs and ANRs, Compositio Math. 39 (1979), no. 2, 129–139. MR 546964
John Milnor, On spaces having the homotopy type of a $\textrm {CW}$-complex, Trans. Amer. Math. Soc. 90 (1959), 272–280. MR 100267, DOI 10.1090/S0002-9947-1959-0100267-4
J. van Mill, Infinite-dimensional topology, North-Holland Mathematical Library, vol. 43, North-Holland Publishing Co., Amsterdam, 1989. Prerequisites and introduction. MR 977744
Katsuro Sakai, The space of cross sections of a bundle, Proc. Amer. Math. Soc. 103 (1988), no. 3, 956–960. MR 947690, DOI 10.1090/S0002-9939-1988-0947690-7
Jaka Smrekar, Compact open topology and CW homotopy type, Topology Appl. 130 (2003), no. 3, 291–304. MR 1978893
J. Smrekar, CW type of inverse limits and function spaces, arXiv:math.AT/07082838.
J. Smrekar, Homotopy type of mapping spaces and existence of geometric exponents, Forum Math., in press.
James Stasheff, A classification theorem for fibre spaces, Topology 2 (1963), 239–246. MR 154286, DOI 10.1016/0040-9383(63)90006-5
H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262. MR 611763, DOI 10.4064/fm-111-3-247-262
H. Toruńczyk, A correction of two papers concerning Hilbert manifolds: “Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds” [Fund. Math. 101 (1978), no. 2, 93–110; MR0518344 (80g:57019)] and “Characterizing Hilbert space topology” [ibid. 111 (1981), no. 3, 247–262; MR0611763 (82i:57016)], Fund. Math. 125 (1985), no. 1, 89–93. MR 813992, DOI 10.4064/fm-125-1-89-93
A. Yamashita, Non-separable Hilbert manifolds of continuous mappings, arXiv:math.GN/ 0610214v1.