scispace - formally typeset
Search or ask a question
Topic

Hilbert cube

About: Hilbert cube is a research topic. Over the lifetime, 428 publications have been published within this topic receiving 4382 citations.


Papers
More filters
Journal ArticleDOI
TL;DR: In this paper, it was shown that the absolute neighborhood retracts (ANR's) are the image of a Q-manifold by a cell-like mapping if and only if whenever A is em-exponential.
Abstract: After Borsuk introduced the notion of absolute neighborhood retracts (ANR's) and J. H. C. Whitehead demonstrated that all ANR's have the homotopy types of cell-complexes [34], the question naturally arose as to whether compact (metric) ANR's must necessarily be homotopy-equivalent to finite cell-complexes. Borsuk expressly posed this conjecture in his address to the Amsterdam Congress in 1954 15], and over the ensuing years considerable progress was made (see [261), including (a) ANR's admitting "brick decompositions," by Borsuk [5], (b) the simply connected case, by de Lyra [21], (c) products with the circle, by M. Mather [22], (d) applications of Wall's obstruction to finiteness [29], (e) compact n-manifolds, by Kirby and Siebenmann [16], and (f) compact Hilbert cube manifolds and locally triangulable spaces, by Chapman [9]. The full problem remained open, however. In this paper the conjecture is settled positively by application of Hilbert cube manifold theory. Specifically, it is shown that each compact ANR is the image of some Hilbert cube manifold (Q-manifold) by a cell-like (CE) mapping. Such mappings, between ANR's, are always homotopy-equivalences [14], [17], [20], [28] (although for more general metric compacta they need not be shape-equivalences [27]), so that by appealing to (f) above, the conjecture is settled. In the process, a considerably stronger result is established, namely, that there exists a cell-like map f from a Q-manifold onto the ANR X whose mapping cylinder M(f) is itself a Q-manif old. The mapping cylinder collapse of M(f) to X provides a particularly nice CE-map of a Q-manifold to the ANR. The general outline of the proof is as follows: First, it is shown that the mapping cylinder of a CE-map from a Q-manifold to an ANR is always a Q-manifold. Second, this result is used to show that the ANR A is the image of a Q-manifold by a CE-mapping if and only if whenever A is em-

139 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the homeomorphism group of a compact Q-manifold is an ANR and that the nerve of a suitably nice open cover of a complex is simple homotopy equivalent to the complex.
Abstract: In this paper, we prove that the homeomorphism group, H(M), of a compact Q-manifold is an ANR. Results of Geoghegan and Torunczyk then show that H(M) is an 12-manif old. As by-products of the proof, we obtain a CE approximation theorem for 12-manifolds, a Vietoris theorem for simple homotopy theory (generalizing the result that a CE map between complexes is simple), and a proof that the nerve of a suitably nice open cover of a complex is simple homotopy equivalent to the complex.

90 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the complement of any compact subset of Hilbert space is contractible, i.e., it is possible to construct a compact zero-dimensional set whose complement is not simply connected.
Abstract: The principal purpose of this paper is to construct in Euclidean space R n, n > 3, a compact zero-dimensional set, A, whose complement is not simply connected. The construction constitutes an affirmative answer to a problem proposed by Borsuk [1], and is a generalization of Antoine's set [2] for the case n = 3. The paper is divided into three parts. In Part 1 the set A is constructed and proved to be zero-dimensional. In Part 2, 7rl(S nA) is computed explicitly and represented non-trivially2 in the symmetric group S6 . In Section 3 it is shown that the construction cannot be accomplished in Hilbert space. More precisely, it is shown that the complement of any compact subset of Hilbert space is contractible. I have been unable to determine whether a set of finite dimension can leave the Hilbert cube multiply connected, but have shown, however, that if there be such a set, then for some n its projection in the subspace xi = x2 = * = Xn = 0 is of infinite dimension. An analogous result is shown to hold for subsets of Rn; namely, the projection of such a set in any (n 1)-dimensional hyperplane has positive dimension. An interesting corollary to this, pointed out to me by E. E. Floyd, is the fact that the set A C Rn C Rn+l leaves Rn+l simply connected. Finally in Section 3, a remark by J. WV. Alexander [4] on Antoine's construction is generalized to show that for each 1 < q < n there is a q-cell in Rn whose complement is not simply connected.

83 citations

Network Information
Related Topics (5)
Homology (mathematics)
9.5K papers, 176.8K citations
83% related
Homotopy
14.8K papers, 267.4K citations
83% related
Metric space
18.4K papers, 351.9K citations
81% related
Automorphism
15.5K papers, 190.6K citations
81% related
Cohomology
21.5K papers, 389.8K citations
79% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
20219
202014
20199
201812
201711