Separable complete ANR's admitting a group structure are Hilbert manifolds

Dobrowolski, T.; Toruńczyk, H.

doi:10.1016/0166-8641(81)90001-8

Cited by 53 publications

(47 citation statements)

References 12 publications

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“…Anderson [5] proved that H ∂ (I) is homeomorphic to the separable Hilbert space 2 (see [8,Proposition VI.8.1] or [32]). It was shown by Luke and Mason [35] that H ∂ (I 2 ) is an absolute retract, which implies that H ∂ (I 2 ) ≈ 2 (apply for instance Dobrowolski and Toruńczyk [26]). For n ≥ 3 it is open whether H ∂ (I n ) is an absolute retract.…”

Section: Introductionmentioning

confidence: 99%

Erdos space and homeomorphism groups of manifolds

Dijkstra

Mill

2010

Memoirs of the AMS

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Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number.Back number information. For back issues see www.ams.org/bookstore. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by payment. Other correspondence should be addressed to 201 Charles Street, Providence, RI 02904-2294 USA.Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to reprint-permission@ams.org. iii Memoirs of the American Mathematical Society AbstractLet M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary countable dense subset of M . Consider the topological group H(M, D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H(M, D) as follows. If M is a one-dimensional topological manifold, then we proved in an earlier paper that H(M, D) is homeomorphic to Q ω , the countable power of the space of rational numbers. In all other cases we find in this paper that H(M, D) is homeomorphic to the famed Erdős space E, which consists of the vectors in Hilbert space 2 with rational coordinates. We obtain the second result by developing topological characterizations of Erdős space. If X is compact then the standard topology on the group of homeomorphisms H(X) of X is the so-called compact-open topology (which coincides with the topology of uniform convergence). For noncompact locally compact spaces we give H(X) the topology that this group inherits from H(αX), where αX is the one-point compactification. In either case we have that H(X) a Polish topological group. If A is a subset of a space X then H(X, A) stands for the subgroup {h ∈ H(X) : h(A) = A} of H(X).Brouwer [11] showed that R is countable dense homogeneous, that is, for all countable dense subsets A and B of R there is an h ∈ H(R) with h(A) = B. It is not difficult to prove that every R n has this property. In view of Brouwer's result it is a natural idea to investigate the group H(R n , Q n ). It was shown in Dijkstra and van Mill [21] that the group H(R, Q) is homeomorphic to the z...

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Section: Introductionmentioning

confidence: 99%

Erdos space and homeomorphism groups of manifolds

Dijkstra

Mill

2010

Memoirs of the AMS

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“…If A is compact, then A is homeomorphic to the Hilbert cube (see [3, p. 100]). If A is not locally compact, then A is homeomorphic to 2 [6]. Therefore, A is invertible in both cases.…”

Section: Corollary 11 Every Non-empty Open Convex Subset Of An Invermentioning

confidence: 99%

“…Let U be a proper open subset of an invertible space X. If U has any of the following properties, then X also has the corresponding properties: (1) T 0 , (2) T 1 , (3) Hausdorff, (4) regular, (5) completely regular, (6) normal, (7) first countable, (8) second countable, (9) separable, (10) metrizable, (11) uniformizable, (12) compact, (13) pseudocompact, (14) extremally disconnected; unless X is a two point space, the list also includes: (15) T 1 and connected, and (16) T 1 and path connected.…”

Section: Theorem 1 (Doyle and Hocking [8]) A Topological N-manifold mentioning

confidence: 99%

Invertibility in infinite-dimensional spaces

Tseng

Wong

1999

Proc. Amer. Math. Soc.

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Abstract. An interesting result of Doyle and Hocking states that a topological n-manifold is invertible if and only if it is a homeomorphic image of the n-sphere S n . We shall prove that the sphere of any infinite-dimensional normed space is invertible. We shall also discuss the invertibility of other infinite-dimensional objects as well as an infinite-dimensional version of the Doyle-Hocking theorem.

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“…In order to verify the discrete approximation property of T(X), we use the following easy modification of [DT,Remark 2] Then X has the discrete approximation property.…”

Section: Preliminariesmentioning

confidence: 99%

The Space of Cross Sections of a Bundle

Sakai¹

1988

Proceedings of the American Mathematical Society

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Separable complete ANR's admitting a group structure are Hilbert manifolds

Cited by 53 publications

References 12 publications

Erdos space and homeomorphism groups of manifolds

Erdos space and homeomorphism groups of manifolds

Invertibility in infinite-dimensional spaces

The Space of Cross Sections of a Bundle

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