## Function spaces of CW homotopy type are Hilbert manifolds.(English)Zbl 1163.54010

Let $$X$$ be a countable CW complex with $$\dim X\geq 1$$, and $$Y$$ be a separable completely metrizable ANR without isolated points. Let $$Y^X$$ denote the space of all continuous mappings from $$X$$ into $$Y$$ with the compact open topology. Let $$\ell^2$$ denote the separable real Hilbert space of square summable sequences. The authors main result says that the following properties are equivalent: (i) $$Y^X$$ is an $$\ell^2$$-manifold; (ii) $$Y^X$$ is an ANR; (iii) $$Y^X$$ has the homotopy type of a CW complex. Several applications are given where further assumptions are imposed on $$X$$ and $$Y$$.

### MSC:

 54C35 Function spaces in general topology 55M15 Absolute neighborhood retracts 57N20 Topology of infinite-dimensional manifolds
Full Text:

### References:

 [1] Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, PWN — Polish Scientific Publishers, Warsaw, 1975. Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58]. · Zbl 0304.57001 [2] Robert Cauty, Une caractérisation des rétractes absolus de voisinage, Fund. Math. 144 (1994), no. 1, 11 – 22 (French, with English summary). · Zbl 0812.54026 [3] Tammo tom Dieck, Partitions of unity in homotopy theory, Composito Math. 23 (1971), 159 – 167. · Zbl 0212.55804 [4] T. Dobrowolski and H. Toruńczyk, Separable complete ANRs admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), no. 3, 229 – 235. · Zbl 0472.57009 [5] Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. · Zbl 0684.54001 [6] László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. · Zbl 0209.05503 [7] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001 [8] Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. · Zbl 0029.32203 [9] 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. · Zbl 0996.54002 [10] C. R. F. Maunder, Algebraic topology, Dover Publications, Inc., Mineola, NY, 1996. Reprint of the 1980 edition. · Zbl 0435.55001 [11] E. Michael, Uniform ARs and ANRs, Compositio Math. 39 (1979), no. 2, 129 – 139. [12] John Milnor, On spaces having the homotopy type of a \?\?-complex, Trans. Amer. Math. Soc. 90 (1959), 272 – 280. · Zbl 0084.39002 [13] J. van Mill, Infinite-dimensional topology, North-Holland Mathematical Library, vol. 43, North-Holland Publishing Co., Amsterdam, 1989. Prerequisites and introduction. · Zbl 0663.57001 [14] Katsuro Sakai, The space of cross sections of a bundle, Proc. Amer. Math. Soc. 103 (1988), no. 3, 956 – 960. · Zbl 0681.58010 [15] Jaka Smrekar, Compact open topology and CW homotopy type, Topology Appl. 130 (2003), no. 3, 291 – 304. · Zbl 1028.55010 [16] J. Smrekar, CW type of inverse limits and function spaces, arXiv:math.AT/07082838. · Zbl 1028.55010 [17] J. Smrekar, Homotopy type of mapping spaces and existence of geometric exponents, Forum Math., in press. · Zbl 1192.55010 [18] James Stasheff, A classification theorem for fibre spaces, Topology 2 (1963), 239 – 246. · Zbl 0123.39705 [19] H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247 – 262. · Zbl 0468.57015 [20] H. Toruńczyk, A correction of two papers concerning Hilbert manifolds: ”Concerning locally homotopy negligible sets and characterization of \?$$_{2}$$-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. [21] A. Yamashita, Non-separable Hilbert manifolds of continuous mappings, arXiv:math.GN/ 0610214v1. · Zbl 1378.57032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.