Homotopy orbits of free loop spaces

Marcel Bökstedt; Iver Ottosen

Displaying similar documents to “Homotopy orbits of free loop spaces”

Dugundji extenders and retracts on generalized ordered spaces

Gary Gruenhage, Yasunao Hattori, Haruto Ohta (1998)

Fundamenta Mathematicae

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For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{c h}$ -extender (resp. $L_{c c h}$ -extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous $L_{c h}$ -extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology...

A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

W. Jurkat, D. Nonnenmacher (1994)

Fundamenta Mathematicae

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Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^{n}$ , we are led to an integration process over quite general sets $A \subseteq q ℝ^{n}$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.

For almost every tent map, the turning point is typical

Henk Bruin (1998)

Fundamenta Mathematicae

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Let $T_{a}$ be the tent map with slope a. Let c be its turning point, and $μ_{a}$ the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, $ʃ g d μ_{a} = l i m_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} g (T_{a}^{i} (c))$ . As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.

From Newton’s method to exotic basins Part I: The parameter space

Krzysztof Barański (1998)

Fundamenta Mathematicae

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This is the first part of the work studying the family $𝔉$ of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of $𝔉$ and give a detailed study of the subfamily $ℱ_{2}$ consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in $ℱ_{2}$ from Newton maps to maps with so-called exotic basins.

An ordinal version of some applications of the classical interpolation theorem

Benoît Bossard (1997)

Fundamenta Mathematicae

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Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space $E_{1}$ with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space $E_{2}$ with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of $E_{1}$ and $E_{2}$ can be controlled by the Szlenk index of E, where the...

Spaces of upper semicontinuous multi-valued functions on complete metric spaces

Katsuro Sakai, Shigenori Uehara (1999)

Fundamenta Mathematicae

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Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by $U S C C_{B} (X)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $φ \in U S C C_{B} (X)$ with its graph which is a closed subset of X × ℝ. The space $U S C C_{B} (X)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $U S C C_{B} (X)$ is homeomorphic...

Linear orders and MA + ¬wKH

Zoran Spasojević (1995)

Fundamenta Mathematicae

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I prove that the statement that “every linear order of size $2^{ω}$ can be embedded in $(ω^{ω}, ≪)$ ” is consistent with MA + ¬ wKH.

Universal spaces in the theory of transfinite dimension, II

Wojciech Olszewski (1994)

Fundamenta Mathematicae

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We construct a family of spaces with “nice” structure which is universal in the class of all compact metrizable spaces of large transfinite dimension $ω_{0}$ , or, equivalently, of small transfinite dimension $ω_{0}$ ; that is, the family consists of compact metrizable spaces whose transfinite dimension is $ω_{0}$ , and every compact metrizable space with transfinite dimension $ω_{0}$ is embeddable in a space of the family. We show that the least possible cardinality of such a universal family is equal to the...

Computing Reidemeister classes

Davide Ferrario (1998)

Fundamenta Mathematicae

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In order to compute the Nielsen number N(f) of a self-map f: X → X, some Reidemeister classes in the fundamental group $π_{1} (X)$ need to be distinguished. In this paper some algebraic results are given which allow distinguishing Reidemeister classes and hence computing the Reidemeister number of some maps. Examples of computations are presented.

A Lefschetz-type coincidence theorem

Peter Saveliev (1999)

Fundamenta Mathematicae

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A Lefschetz-type coincidence theorem for two maps f,g: X → Y from an arbitrary topological space to a manifold is given: $I_{f g} = λ_{f g}$ , that is, the coincidence index is equal to the Lefschetz number. It follows that if $λ_{f g} \neq 0$ then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains well-known coincidence results for (i) X,Y manifolds, f boundary-preserving, and (ii) Y Euclidean, f with acyclic fibres. It also implies certain fixed point results for multivalued maps with “point-like”...

The minimum uniform compactification of a metric space

R. Grant Woods (1995)

Fundamenta Mathematicae

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It is shown that associated with each metric space (X,d) there is a compactification $u_{d} X$ of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of $u_{d} X$ are presented, and a detailed study of the structure of $u_{d} X$ is undertaken. This culminates in a topological characterization of the outgrowth $u_{d} ℝ^{n} ∖ ℝ^{n}$ , where $(ℝ^{n}, d)$ is Euclidean n-space with its usual metric. ...

The space of ANR’s in $ℝ^{n}$

Tadeusz Dobrowolski, Leonard Rubin (1994)

Fundamenta Mathematicae

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The hyperspaces $A N R (ℝ^{n})$ and $A R (ℝ^{n})$ in $2^{ℝ^{n}} (n \geq 3)$ consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute $G_{δ σ δ}$ -spaces and that, indeed, they are not $F_{σ δ σ}$ -spaces. The main result is that $A N R (ℝ^{n})$ is an absorber for the class of all absolute $G_{δ σ δ}$ -spaces and is therefore homeomorphic to the standard model space $Ω_{3}$ of this class.

Cantor manifolds in the theory of transfinite dimension

Wojciech Olszewski (1994)

Fundamenta Mathematicae

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For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space $Z_{α}$ such that $i n d Z_{α} = α$ , and no closed subset L of $Z_{α}$ with ind L less than the predecessor of α is a partition in $Z_{α}$ . An α-dimensional Cantor Ind-manifold can be constructed similarly.

A partial order where all monotone maps are definable

Martin Goldstern, Saharon Shelah (1997)

Fundamenta Mathematicae

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It is consistent that there is a partial order (P,≤) of size $ℵ_{1}$ such that every monotone function f:P → P is first order definable in (P,≤).

A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

Witold Marciszewski (1997)

Fundamenta Mathematicae

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We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions $c_{p} (X)$ onto $c_{p} (X)$ × ℝ $. I n p a r t i c u l a r,$ cp(X) $i s n o t l i n e a r l y h o m e o m o r p h i c t o$ cp(X) $\times ℝ$ . One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

Categoricity of theories in Lκω , when κ is a measurable cardinal. Part 1

Saharon Shelah, Oren Kolman (1996)

Fundamenta Mathematicae

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We assume a theory T in the logic $L_{κ ω}$ is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.

The Zahorski theorem is valid in Gevrey classes

Jean Schmets, Manuel Valdivia (1996)

Fundamenta Mathematicae

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Let Ω,F,G be a partition of $ℝ^{n}$ such that Ω is open, F is $F_{σ}$ and of the first category, and G is $G_{δ}$ . We prove that, for every γ ∈ ]1,∞[, there is an element of the Gevrey class Γγ which is analytic on Ω, has F as its set of defect points and has G as its set of divergence points.

Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

Alessandro Andretta, Alberto Marcone (1997)

Fundamenta Mathematicae

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We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $\prod_{2}^{0}$ -complete and that the set of Cauchy problems which locally have a unique solution is $\sum_{3}^{0}$ -complete. We prove that the set of Cauchy problems which have a global solution is...