Space Of Homeomorphisms Research Articles

On chain continuity

A number of recent papers examine for a dynamical system $f: X\rightarrow X$ the concept of equicontinuity at a point. A point$x \in X$ is an equicontinuity point for $f$ if for every$\epsilon > 0$ there is a $\delta > 0$ so that the orbit ofinitial points $\delta$ close to $x$ remains at all times$\epsilon$ close to the corresponding points of the orbit of $x$,i.e. $d(x,x_0) 0$ there is a$\delta > 0$ so that all $\delta$ chains beginning $\delta$ closeto $x$ remain $\epsilon$ close to the points of the orbit of $x$,i.e. $d(x,x_0) < \delta$ and $d(f(x_i),x_{i+1}) \leq \delta$imply $d(f^i(x),x_i) \leq \epsilon$ for $i = 1,2,\ldots$. Inthis note we characterize this property of chain continuity.Despite the strength of this property, there is a class ofsystems $(X,f)$ for which the chain continuity points form aresidual subset of the space $X$. For a manifold $X$ this classincludes a residual subset of the space of homeomorphisms on $X$.

Typical Transitivity for Lifts of Rotationless Annulus or Torus Homeomorphisms

We say that a homeomorphism h of the base space X (which may be either the annulus or n-torus, n⩾2) is rotationless if it is area-preserving and has a lift h∼ to the covering space X∼([0,1] × R or Rn) with mean translation zero (∫Ω(h∼(x)–x)dx=0, where Ω is [0,1] × [0,1]). We prove (Theorem 1) that in the space of rotationless homeomorphisms of X with the uniform topology, the subspace consisting of homeo-morphisms with transitive lifts to X ∼ contains a dense Gδ subset. This extends our earlier result, valid only when the base space is the annulus, that typical rotationless homeomorphisms have recurrent lifts. Our result also extends that of Besicovitch, who in 1937 exhibited the first transitive homeomorphism of the plane. In this context we establish such a homeomorphism which is additionally spatially periodic.

On the dimension of certain totally disconnected spaces

It is well known that there exist separable, metrizable, totally disconnected spaces of all dimensions. In this note we introduce the notion of an almost 0-dimensional space and prove that every such space is a totally disconnected subspace of an R-tree and, hence, at most 1-dimensional. As applications we prove that the spaces of homeomorphisms of the universal Menger continua are 1-dimensional and that hereditarily locally connected spaces have dimension at most two.

Some non-uniformly homeomorphic spaces

in this paper we prove that for 0<p, q≤1 the real F-spacesL q[0, 1] and l p are not uniformly homeomorphic. The particular casep=q=1 is due to Enflo and our work is motivated by his.

On spaces of equivariant isomorphisms

If π: X→ X/G is a covering projection, where G is a group of homeomorphisms (diffeomorphisms) acting freely on X, one can talk of spaces of π-equivariant homeomorphisms (diffeomorphisms). The study of such spaces is of interest to equivariant surgery. In this paper, we obtain certain results about these spaces, which in turn, allow us to compute the 0-homotopy groups of them in the case where π is the obvious map R n → T n , for n⩾5.

An embedding space triple of the unit interval into a graph and its bundle structure

Let l 2 denote a Hilbert space, and let l 2 Q ={(x i )∈l 2 |sup|i.x i |<∞} and l 2 f ={(x i )∈l 2 |x i =0 except for finitely many i}. We show that the triple (H(X), H LIP (X), H PL (X)) of spaces of homeomorphisms, of Lipschitz homeomorphisms, and of PL homeomorphisms of a finite graph X onto itself is an (l 2 ,L 2 Q , l 2 f )-manifold triple, and that the triple (E(I,X), E LIP (I,X), E PL 5I,X)) of spaces of embeddings, of Lipschitz embeddings, and of PL embeddings of I=[0,1] into a graph X is an (l 2 ,l 2 Q ,l 2 4 )-manifold triple

On the space of Lipschitz homeomorphisms of a compact polyhedron

Let X be a positive dimensional compact Euclidean polyhedron. Let H(X), HUP{X) and HPL(X) be respectively the space of homeomorphisms, the space of Lipschitz homeomorphisms and the space of piecewise-lin ear homeomorphisms of X onto itself. In this paper, we establish a homeomorphism taking the triple (H(X), HUp(X), H?L(X)) onto the triple (H(X) x s, HUp(X) x Σ, HFL(X) X σ), where 5 = (-1, l)ω, Σ = {(*,) e s\sup\Xi < 1} and σ = {(*,) e s\x t = 0 except for finitely many /}. As a consequence we prove that when X is a PL manifold with dim c Φ 4 and dX = 0, in case dimZ = 5, (H(X),HUP(X)) is an (s, Σ)-manifold pair if H(X) is an s-manifold. We also prove that if dim* = 1 or 2, then (H(X),HPL(X)) is an (5, σ)-manifold pair and (H(X),HUp(X)) is an (s, Σ)-manifold.

RATIONAL HERMITIAN K-THEORY AND DIHEDRAL HOMOLOGY

This paper studies the relations between Hermitian K-theory of topological spaces and the dihedral homology of differential graded algebras with involution. The results are applied to describe the rational homotopy type of the space of homeomorphisms of a simply-connected compact manifold. Bibliography: 25 titles.

Compactifying the space of homeomorphisms

Compactifying the space of homeomorphisms

On the space of linear homeomorphisms of a polyhedral N-cell

The author shows that the space of all the linear homeomorphisms of a triangulated n-cell into the n-dimensional euclidean space E n has the structure of a fibre bundle with an open n-cell as its fibres. If the triangulation has exactly two interior vertices, the fibre bundle is homeomorphic to the product bundle E n × E n . This improves a previous result of the author, which was obtained in answering a question raised by R.H. Bing and M. Starbird.

An almost uniquely homogeneous subgroup of [formula omitted

An example is constructed of a connected locally connected ( n–1)-dimensional subgroup X of R with the property that for every homeomorphism g in H( X) there is an h ϵ X such that g( x)= x+ h or g( x)=- x+ h. The dimension of the space of homeomorphisms H( X) will also be n-1. A new proof of a theorem of Barit and Renaud is also given. In particular, let X be a separable metric space which is either compact or locally connected. If X is uniquely homogeneous, then X is trivial or the group of order two.

An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p n}, 1≦p n<∞,p n→p, the spaceX=(Σ⊕l p n) q (resp.U=(Σ⊕L p n(0, 1)) q ) is uniformly homeomorphic toX⊕l p (resp.U⊕L p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.

Expressive power in first order topology

AbstractAfirst order representation(f.o.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions “one f.o.r. is at least as expressive as another relative to a class of spaces” and “one class of spaces is definable in another relative to an f.o.r.”, and prove some general statements. Following this we compare some well-known classes of spaces and first order representations. A principal result is that ifXandYare two Tichonov spaces whose posets of zero-sets are elementarily equivalent then their respective rings of bounded continuous real-valued functions satisfy the same positive-universal sentences. The proof of this uses the technique of constructing ultraproducts as direct limits of products in a category theoretic setting.

Completely Metrizable Spaces of Embeddings

A study is made of the complete metrizability of spaces of embeddings and spaces of homeomorphisms under the compact-open topology. It is shown that the space of continuous injections from a hemicompact metric space into a complete metric space is completely metrizable.

Completely metrizable spaces of embeddings

A study is made of the complete metrizability of spaces of embeddings and spaces of homeomorphisms under the compact-open topology. It is shown that the space of continuous injections from a hemicompact metric space into a complete metric space is completely metrizable.

Approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections

In this paper, we prove that approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections. The principal tools are the torus trick developed by Kirby and Siebenmann, a surgery theorem concerning homotopy structures on torii due to Hsiang and Wall, a theorem on the space of homeomorphisms of the torus due to Hamstrom and a generalization of hereditary homotopy equivalence developed by the author.

The closure of the space of homeomorphisms on a manifold. The piecewise linear case

The closure of the space of homeomorphisms on a manifold. The piecewise linear case

A near-selection theorem

A near-selection theorem is proven for carriers defined on spaces that are the countable union of finite dimensional compacta. As an application a new proof is given of the fact that the space of homeomorphisms on a compact piecewise linear manifold is locally contractible. In addition a new criterion is given to determine if the space of homeomorphisms on a compact n-manifold is an l 2-manifold.

On the space of piecewise linear homeomorphisms of a manifold

Let M n {M^n} be a compact PL manifold, n ≠ 4 n \ne 4 ; if n = 5 n = 5 , suppose ∂ M \partial M is empty. Let H ( M ) H(M) be the space of homeomorphisms on M M and H ∗ ( M ) {H^{\ast }}(M) the elements of H ( M ) H(M) which are isotopic to PL homeomorphisms. It is shown that the space of PL homeomorphisms, P L H ( M ) PLH(M) , has the finite dimensional compact absorption property in H ∗ ( M ) {H^{\ast }}(M) and hence that ( H ∗ ( M ) , P L H ( M ) ) ({H^{\ast }}(M),PLH(M)) is an ( l 2 , l 2 f ) ({l_2},l_2^f) -manifold pair if and only if H ( M ) H(M) is an l 2 {l_2} -manifold. In particular, if M 2 {M^2} is a 2 2 -manifold, ( H ( M 2 ) , P L H ( M 2 ) ) (H({M^2}),PLH({M^2})) is an ( l 2 , l 2 f ) ({l_2},l_2^f) -manifold pair.

Topological description of the space of homeomorphisms on closed $2$-manifolds

Topological description of the space of homeomorphisms on closed $2$-manifolds