Compact Hilbert Cube Manifold Research Articles

Chain recurrent sets of generic mappings on compact spaces

Let 0-CR denote the class of all metric compacta X such that the set of maps f:X→X with 0-dimensional sets CR(f) of chain recurrent points is a dense Gδ-subset of the mapping space C(X,X) (with the uniform convergence). We prove, among others, that countable products of polyhedra or locally connected curves belong to 0-CR. Compacta that admit, for each ϵ>0, an ϵ-retraction onto a subspace from 0-CR belong to 0-CR themselves. Perfect ANR-compacta or n-dimensional LCn−1-compacta have perfect CR(f) for a generic self-map f. In the cases of polyhedra, compact Hilbert cube manifolds, local dendrites and their finite products, a generic f has CR(f) being a Cantor set and the set of periodic points of f of arbitrarily large periods is dense in CR(f). The results extend some known facts about CR(f) of generic self-maps f on PL-manifolds.

Delooping controlled pseudo-isotopies of Hilbert cube manifolds

This paper is concerned with the controlled simple homotopy theory and controlled pseudo-isotopy theory of a Hurewicz fibration p: E→ B from a compact Hilbert cube manifold E to a compact polyhedron B. The main result is that the controlled pseudo-isotopy space is homotopy equivalent to the loop space of the controlled Whitehead space.

Bounded homotopy equivalences of Hilbert cube manifolds

Let M and F be Hubert cube manifolds with F compact. The purpose of this paper is to study homotopy equivalences/: M -» R' X F which have bounded control in the Redirection. Roughly, these homotopy equivalences form a semi-sim- plicial complex #^4(Rm X F), the controlled Whitehead space. Using results about approximate fibrations, x F ) is related to the semi-simplicial complex of bounded concordances on R' X F. Then the homotopy groups of #^(Rm X F) are computed in terms of the lower algebraic AT-theoretic functors K_i. 1. Introduction. Let F be a compact Hilbert cube manifold. We are interested in homotopy equivalences /: M -» Rm X F which are controlled in the Redirection, where M is also a Hilbert cube manifold. To say / is controlled in the Redirection means that pf: M -» Rm is an approximate fibration, where p: Rm X F -* Rm is projection. The collection of all such homotopy equivalences, which are additionally given to be retractions onto the collared submanifold Rm X F of M, form the vertices of a semi-simplicial complex ?TA(Rm X F) (see §2 for the precise definition). The main result of this paper is the computation of the homotopy groups of WA(Rm X F) (see Corollary 1 below). In order to do this we relate WA(Rm X F) to the semi-simplicial complex ^h(Rm X F) of bounded concordances on Rm X F. An «-simplex of <eb^Rm X F) is a homeomorphism

Approximate fibrations and bundle maps on Hilbert cube manifolds

This paper is concerned with parameterized families of approximate fibrations from a compact Hilbert cube manifold M to a compact polyhedron B. The main result shows how to straighten out certain of these families to be nearly like a product. As an application of this technique, it is shown that an approximate fibration p: M → B can be approximated arbitrarily closely by bundle maps if and only if p is homotopic via approximate fibrations to a bundle map. Another result is that the space of bundle maps from M to B is locally n-connected for each n ⩾ 0.

Mapping Hilbert Cube Manifolds to ANR's: A Solution of a Conjecture of Borsuk

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-

The Homeomorphism Group of a Compact Hilbert Cube Manifold is an ANR

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.

Applications of group actions on finite complexes to Hilbert cube manifolds

For any compact Hilbert cube manifold M such that H ~ ∗ ( M , Z p ) {\tilde H_\ast }(M,{Z_p}) , there exists an embedding g of M into the Hilbert cube Q such that g ( M ) g(M) is the fixed point set of a semifree periodic homeomorphism of Q with period p. A counterexample is given to the conjecture that any two proper homotopic period p homeomorphisms of a Hilbert cube manifold such that the homeomorphisms revolve trivially about a unique fixed point are equivalent. A counterexample is also given for the case where the fixed point set is empty.

Concordances of Hilbert cube manifolds

The main result of this paper asserts that homotopy groups of concordances of compact Hilbert cube manifolds are isomorphic to homotopy groups of concordances of compact finite-dimensional piecewise-linear manifolds. This enables us to apply some finite-dimensional results to obtain some new information about homotopy groups of homeomorphism groups of compact Hilbert cube manifolds. Our approach also yields a much shorter proof of the local contractibility of the homeomorphism group of any compact Hilbert cube manifold.

Mapping cylinders of Hilbert cube factors II. the relative case

A concept of relative or reduced mapping cylinders is introduced, and it is shown that if ƒ is any map between two spaces which have the property that their products with the Hilbert cube are compact Hilbert cube manifolds, then for any closed subset A of the domain of ƒ the relative mapping cylinder of ƒ reduced modulo A also has this property and that the two collapses associated with it generate in a natural way near-homeomorphisms of Hilbert cube manifolds.

Compact Hilbert cube manifolds and the invariance of Whitehead torsion

Compact Hilbert cube manifolds and the invariance of Whitehead torsion