On Homeomorphisms of Infinite-Dimensional Bundles. I

Wong, Raymond Y. T.

doi:10.2307/1996993

Cited by 6 publications

(7 citation statements)

References 3 publications

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“…In the following we say 11 is a normal open cover of X\X, where K is closed in the metric space X, provided each map /: X\K -» X which is limited by 11 has an extension /: X -» X which is the identity on K. normal open cover of E\H refining 11 and let /(llj) denote the induced cover of E. Using Theorem 3.1 of [7] there is an isomorphism g of F such that gfÍK\H)° (U">0 AT")) = 0 and g is limited by 0A. Then f lg{: e\h -* e\h is a B-preserving homeomorphism which rè limited by 11. and satisfies f~ gfiK\H)C\ [(J">0 ÍKn\H)] = 0.…”

Section: T a Chapman And R Y T Wongmentioning

confidence: 99%

“…Let K be given by the first half of the theorem. If E is the product bundle (B x M, p, B) we can apply Theorem 2 of this paper and Lemma 5.1 of [7] to obtain a strong extraction of K from E. In general we can, by hypothesis of and any open cover ll of U for which G is limited, we may choose \hf\ to be a (F, St4(H))-zso/opy.…”

Section: T a Chapman And R Y T Wongmentioning

confidence: 99%

“…In [7] several aspects of homeomorphism theory are studied in the setting of (fibre) bundles with separable infinite dimensional spaces (manifolds) as fibres. Major results are established for bundles having fibre the Hilbert space l2, or, equivalently, s, the countable infinite product of reals.…”

Section: Introductionmentioning

confidence: 99%

“…Our notation and definition follow that of [7]. In this paper we say a closed subset K of the total space E of a bundle (E, p, B) is a fibre Z-set provided K np'Kb) is a Z-set in p_1(&) for each b £ B. Fibre bundle (E, p, B) will be denoted by its total space E. For any KCE, a map /: K -► E is B-preserving (or fibre-preserving) if pfix) = pix) tot all x £ K.…”

Section: Introductionmentioning

confidence: 99%

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On homeomorphisms of infinite dimensional bundles. II

Chapman¹,

Wong²

1974

Trans. Amer. Math. Soc.

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ABSTRACT. This paper presents some aspects of homeomorphism theory in the setting of (fibre) bundles modeled on separable Hilbert manifolds and generalizes results previously established.The main result gives a characterization of subsets of infinite deficiency in a bundle by means of their restriction to the fibres, from which we are able to prove theorems of the following types: (a) mapping replacement, (b) separation of sets, (c) negligibility of subsets, and (d) extending homeomorphisms.

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Section: T a Chapman And R Y T Wongmentioning

confidence: 99%

Section: T a Chapman And R Y T Wongmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On homeomorphisms of infinite dimensional bundles. II

Chapman¹,

Wong²

1974

Trans. Amer. Math. Soc.

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“…In this paper we continue the study of homeomorphisms and prove an analogue of the homeomorphism extension theorem for bundles modeled on Hilbert cube manifolds; thus we generalize previous results for Ç-manifolds Hypothesis. Throughout the following let M denote a connected Q-manifold and, for matter of convenience, assume that all spaces considered are metrizable.Our notation and definitions follow that of [7] and [8].2. Separation of sets.…”

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confidence: 99%

On Homeomorphisms of Infinite Dimensional Bundles. III

Chapman¹,

Wong²

1974

Transactions of the American Mathematical Society

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ABSTRACT. In this paper we continue the study of homeomorphisms and prove an analogue of the homeomorphism extension theorem for bundles modeled on Hilbert cube manifolds; thus we generalize previous results for Ç-manifolds Hypothesis. Throughout the following let M denote a connected Q-manifold and, for matter of convenience, assume that all spaces considered are metrizable.Our notation and definitions follow that of [7] and [8].2. Separation of sets. We recall that for a bundle E = (E, p, B) and for a pair (F, U) denoting an open set of E and an open cover of F, a (F, ll)-z'soropy ifil on E is a map p = {¡lA: E x I -* E such that pQ -id, each pt is an isomorphism with support in F, and i^.lyl is limited by ll. A closed set K in the product space X x Y is an X-projective Z-set provided the projection of K into X is contained in a closed Z-set of X. A closed subset K of E is a fibre Zset provided K O p~lib) is a Z-set in p~lib) for each b e B. The purpose of this section is to prove the following theorem.

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On homeomorphism spaces of Hilbert manifolds

Wong¹

1983

Proc. Amer. Math. Soc.

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Abstract. Let M be a Hubert manifold modeled on the separable Hubert space l2.We prove the following Fibred Homeomorphism Extension Theorem: Let M X B" -B" be the product bundle over the n-ball B", then any (fibred) homeomorphism on M X S"~ ' extends to a homeomorphism on all M X B" if and only if it extends to a mapping on all M X B". Moreover, the size of the extension may be restricted by any open cover on M X B". The result is then applied to study the space of homeomorphisms %(M) under various topologies given on %(M). For instance, if M = l2 and %(l2) has the compact-open topology, the %(l2) is an absolute extensor for all metric spaces. A counterexample is provided to show that the statement above may not be generalized to arbitrary manifold M. [To] showed that %(M) is, in fact, an ANR.Let M be a separable Hubert manifold modeled on the Hubert space l2. The function space, G(M), can be equipped with the c-o or majorant topology. Here the topology plays an important role since %(M) may fail to be locally contractible in the c-o topology. The following example illustrates that point. Example 1. We remove from the open interval (0,1) a sequence of disjoint open intervals which have their diameters tend to zero as they approach 1. Denote the resulting space by X. Let Y denote the space obtained by attaching circles to the end points in X as illustrated in the figure below. According to [We], M = Y X l2 is a Hubert manifold. But, by switching the upper and lower half of smaller and smaller

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On Homeomorphisms of Infinite-Dimensional Bundles. I

Abstract: ABSTRACT. In this paper we present several aspects of homeomorphism theory in the setting of fibre bundles modeled on separable infinite-dimensional

Cited by 6 publications

References 3 publications

On homeomorphisms of infinite dimensional bundles. II

On homeomorphisms of infinite dimensional bundles. II

On Homeomorphisms of Infinite Dimensional Bundles. III

On homeomorphism spaces of Hilbert manifolds

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