A short proof of Toruńczyk’s characterization theorems

Dijkstra, Jan; Levin, Michael; van Mill, Jan

doi:10.1090/proc/13262

A short proof of Toruńczyk’s characterization theorems
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by Jan J. Dijkstra, Michael Levin and Jan van Mill PDF

Proc. Amer. Math. Soc. 145 (2017), 901-914 Request permission

Abstract:

We present short proofs of Toruńczyk’s well-known characterization theorems of the Hilbert cube and Hilbert space, respectively.

References

S. M. Ageev, Axiomatic method of partitions in the theory of Nöbeling spaces. II. An unknotting theorem, Mat. Sb. 198 (2007), no. 5, 3–32 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 5-6, 597–625. MR 2354284, DOI 10.1070/SM2007v198n05ABEH003851
S. M. Ageev, The axiomatic partition method in the theory of Nöbeling spaces. I. Improving partition connectivity, Mat. Sb. 198 (2007), no. 3, 3–50 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 3-4, 299–342. MR 2354278, DOI 10.1070/SM2007v198n03ABEH003838
S. M. Ageev, Axiomatic method of partitions in the theory of Nöbeling spaces. III. Consistency of the system of axioms, Mat. Sb. 198 (2007), no. 7, 3–30 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 7-8, 909–934. MR 2354531, DOI 10.1070/SM2007v198n07ABEH003866
R. D. Anderson and John D. McCharen, On extending homeomorphisms to Fréchet manifolds, Proc. Amer. Math. Soc. 25 (1970), 283–289. MR 258064, DOI 10.1090/S0002-9939-1970-0258064-5
R. D. Anderson and T. A. Chapman, Extending homeomorphisms to Hilbert cube manifolds, Pacific J. Math. 38 (1971), 281–293. MR 319204
Mladen Bestvina, Characterizing $k$-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (1988), no. 380, vi+110. MR 920964, DOI 10.1090/memo/0380
Mladen Bestvina, Philip Bowers, Jerzy Mogilski, and John Walsh, Characterization of Hilbert space manifolds revisited, Topology Appl. 24 (1986), no. 1-3, 53–69. Special volume in honor of R. H. Bing (1914–1986). MR 872478, DOI 10.1016/0166-8641(86)90049-0
Philip L. Bowers, Dense embeddings of sigma-compact, nowhere locally compact metric spaces, Proc. Amer. Math. Soc. 95 (1985), no. 1, 123–130. MR 796460, DOI 10.1090/S0002-9939-1985-0796460-9
T. A. Chapman, On the structure of Hilbert cube manifolds, Compositio Math. 24 (1972), 329–353. MR 305432
T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR 0423357
A. Chigogidze, Inverse spectra, North-Holland Mathematical Library, vol. 53, North-Holland Publishing Co., Amsterdam, 1996. MR 1406565
Robert D. Edwards, Characterizing infinite-dimensional manifolds topologically (after Henryk Toruńczyk), Séminaire Bourbaki (1978/79), Lecture Notes in Math., vol. 770, Springer, Berlin-New York, 1980, pp. Exp. No. 540, pp. 278–302. MR 572429
William E. Haver, Mappings between $\textrm {ANR}$s that are fine homotopy equivalences, Pacific J. Math. 58 (1975), no. 2, 457–461. MR 385865
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
Michael Levin, A $Z$-set unknotting theorem for Nöbeling spaces, Fund. Math. 202 (2009), no. 1, 1–41. MR 2457483, DOI 10.4064/fm202-1-1
M. Levin, Characterizing Nöbeling spaces, (2006), arXiv:math/0602361.
J. van Mill, Infinite-dimensional topology, North-Holland Mathematical Library, vol. 43, North-Holland Publishing Co., Amsterdam, 1989. Prerequisites and introduction. MR 977744
R. T. Miller, Mapping cylinder neighborhoods of some ANR’s, Ann. of Math. (2) 103 (1976), no. 2, 417–427. MR 402757, DOI 10.2307/1970946
Andrzej Nagórko, Characterization and topological rigidity of Nöbeling manifolds, Mem. Amer. Math. Soc. 223 (2013), no. 1048, viii+92. MR 3088246, DOI 10.1090/S0065-9266-2012-00643-5
H. Toruńczyk, On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), no. 1, 31–40. MR 585543, DOI 10.4064/fm-106-1-31-40
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
John J. Walsh, Characterization of Hilbert cube manifolds: an alternate proof, Geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986, pp. 153–160. MR 925862
James E. West, Mapping Hilbert cube manifolds to ANR’s: a solution of a conjecture of Borsuk, Ann. of Math. (2) 106 (1977), no. 1, 1–18. MR 451247, DOI 10.2307/1971155