Publisher Summary This chapter discusses open problems in infinite-dimensional topology. The development of infinite-dimensional topology was greatly stimulated by three famous open problem lists—that of Geoghegan, West and Dobrowolski, Mogilski. It is now expected that the future progress will happen on the intersection of infinite-dimensional topology with neighbor areas of mathematics—Dimension Theory, Descriptive Set Theory, Analysis, and Theory of Retracts. This chapter attempts to select the problems whose solution would require creating new methods. It defines a pair as a pair (X, Y) consisting of a space X and a subspace Y ⊂ X. The ω denotes the set of non-negative integers. Many results and objects of infinite-dimensional topology have zero-dimensional counterparts usually considered in Descriptive Set Theory. As a rule, “zero dimensional” results have simpler proofs compared to their higher dimensional counterparts. Some zero-dimensional results are proved by essentially zero-dimensional methods (like those of infinite game theory), and it is an open question to which extent their higher-dimensional analogues are true. The chapter elaborates about the higher dimensional descriptive set theory, concepts of Zn-sets, along with presenting the related questions..

Open problems in infinite-dimensional topology

/pdf/monotone-mappings-of-a-two-disk-onto-itself-which-fix-the-hql16cw5e8.pdf

Monotone mappings of a two-disk onto itself which fix the disk's boundary can be canonically approximated by homeomorphisms

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1240433446

Embeddings of infinite-dimensional manifold pairs and remarks on stability and deficiency

The space, H(M), of all mappings of the compact manifold M onto itself which can be approximated arbitrarily closely by homeomorphisms is studied. It is shown that H(M) is homogeneous and weakly locally contractible. If M is a compact 2-manifold without boundary, then H(M) is shown to be locally contractible. 1. Let M be a compact manifold and H(M) denote the space of all homeomorphisms of M onto itself. We shall study the space, H(M), of all continuous functions of M onto itself which can be approximated arbitrarily closely by elements of H(M). All function spaces on compact spaces will be assumed to have the supremum metric,p; i.e., if X and Y are spaces with d the metric on Y and / and g are functions from X into Y, then p(f, g) supx,xtd(f(x), g(x))I. Since M is compact, the topology thus generated agrees with the compact-open topology. A mapping of an n-manifold, Mn, onto itself is said to be cellular if for each y e Mn, f 1(y) can be expressed as the intersection of a nested sequence of ncells. Armentrout (n 5) [20] have recently shown that H(Mn), n g 4, is precisely the space of all cellular mappings of Mn onto itself. Hence most of the results of this paper could be stated in terms of spaces of cellular mappings. Cellular mappings have been studied extensively (cf., Lacher [15], [16]). Let H8(M) denote the space of all homeomorphisms of M onto itself which equal the identity when restricted to the boundary of M and, following our previous notation, let H11(M) denote the space of all continuous functions of M onto itself which can be approximated arbitrarily closely by elements of H8(M). We shall state some of the major results concerning H(M) and H8(M) and then indicate which of the analogous theorems can be proven for H(M) and H18(M): Received by the editors December 20, 1972 and, in revised form, July 9, 1973. AMS (MOS) subject classifications (1970). Primary 54H15, 57E05; Secondary 57A60, 57A20.

/pdf/the-closure-of-the-space-of-homeomorphisms-on-a-manifold-447bvxkxc5.pdf

The closure of the space of homeomorphisms on a manifold

In this paper, we prove that any locally trivialfibre bundles p : X―>B with fibre M a manifold modeled on an infinitedimensional space E (e. g. the Hilbert space l2 or the Hilbert cube Q) is bundle isomorphic to the bundle />°proj: Xx E―>B. Further, we can obtain a strong version of this Bundle StabilityTheorem. From Bundle Stability Theorem, we can introduce the notion of deficiency in bundles. We show that a finite union of locally deficientsets is deficientand we prove a bundle version of Mapping Replacement Theorem. §0. Introduction. A Hilbert (Hilbert cube) manifold, brieflyl2-manifold (Q-manifold), is a paracompact space M admitting an open cover by sets homeomorphic (= ) to open subsets of the Hilbert space lz (the Hilbert cube Q). These manifolds are topologically stable, that is, M^Mxl2 (M^MxQ). This Stability Theorem due to R. D. Anderson and R. M. Schori [A-S] is most fundamental in the theory of infinite-dimensional manifolds. In this paper, we establish the stability theorem for locally trivial fibre bundles with fibre an /2-manifold or a Q-manifold. We will callthese bundles l2-manifold bundles or Q-manifold bundles, respectively. Bundle Stability Theorem. {Assume B is metrizable.) (A) An l2-manifold bundle is bundle isomorphic to p°pro'}:Xxlz-^B. (B) A Q-manifold bundle is bundle isomorphic to £°proj:XxQ―*B. Here a bundle p : X-^B is bundle isomorphic to a bundle p': X'-^B if there exists homeomorphism h : X-+X' such that p'h = p (such a homeomorphism is called a bundle homeomorphism). In this theorem, there is a bundle homeomorphism h : Xxl2-*X (h : XxQ -X) is homotopic to the projection proj: Xxl2-^X (proj: XxQ―*X) by a small bundle Received October 8, 1980.

/pdf/stability-for-infinite-dimensional-fibre-bundles-4f43yr6mj6.pdf

Stability for infinite-dimensional fibre bundles

Stable Function Spaces

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Open problems in infinite-dimensional topology

Monotone mappings of a two-disk onto itself which fix the disk's boundary can be canonically approximated by homeomorphisms

Embeddings of infinite-dimensional manifold pairs and remarks on stability and deficiency

The closure of the space of homeomorphisms on a manifold

Stability for infinite-dimensional fibre bundles

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