Equiconnected spaces and Baire classification of separately continuous functions and their analogs

We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.     

Points of joint continuity and large oscillations

For topological spaces X and Y and a metric space Z, we introduce a new class N( X ×Y, Z ) \mathcal{N}\left( {X \times Y,\,Z} \right) of mappings f: X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping f from this class and any countable-type set B in Y, the set C _B (f) of all points x from X such that f is jointly continuous at any point of the set {x} × B is residual in X: We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and f ? N( X ×Y, Z ) f \in \mathcal{N}\left( {X \times Y,\,Z} \right) , then, for any ε > 0, the projection of the set D ^ε (f) of all points p ∈ X × Y at which the oscillation ω _f (p) ≥ ε onto X is a closed set nowhere dense in X.     

Joint continuity of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript><Emphasis Type="Italic">C</Emphasis>-functions with values in moore spaces

We introduce the notion of categorical cliquish mapping and show that, for each K _h C-mapping f: X × Y → Z, where X is a topological space, Y is a space with the first axiom of countability, and Z is a Moore space, with categorical-cliquish horizontal y-sections f _y , the sets C _y (f) are residual G _δ-type sets in X for every y ∈ Y. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1539–1547, November, 2008.     

Homotopy invariants of mappings to the circle

Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and I _r (a): (X × T) _r → \mathbbZ \mathbb{Z} is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence I _r (a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of Z-valued functions on (X × T) ^r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.     

Order of a function on the Bruschlinsky group of a two-dimensional polyhedron

Homotopy classes of mappings of a compact polyhedron X to the circle T form an Abelian group B(X), which is called the Bruschlinsky group and is cananically isomorphic to H ¹ (X; ℤ), Let L be an Abelian group, and let f: B(X) → L be a function. One says that the order of f does not exceed r if for each mapping a: X → T the value f([a]) is ℤ-linearly expressed via the characteristic function I _r (a): (X × T) ^r → ℤ of (Γ _a ) ^r , where Γ _a ⊂ X × T is the graph of a. The (algebraic) degree of f is not greater than r if the finite differences of f of order r + 1 vanish. Conjecturally, the order of f is equal to the algebraic degree of f. The conjecture is proved in the case where dim X ≤ 2. Bibliography: 1 title.     

Polyhedral banach spaces and extensions of compact operators

LetX be a polyhedral Banach space whose dual is anL ₁(μ) space for some measureμ. Then for each Banach spacesY ⊆Z and each compact operatorT: Y →X there exists a norm preserving compact extension Z →X.     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Joint Continuity and Quasicontinuity of Horizontally Quasicontinuous Mappings

We show that if Xis a topological space, Ysatisfies the second axiom of countability, and Zis a metrizable space, then, for every mapping f: X× Y Zthat is horizontally quasicontinuous and continuous in the second variable, a set of points x Xsuch that fis continuous at every point from {x} × Yis residual in X. We also generalize a result of Martin concerning the quasicontinuity of separately quasicontinuous mappings.     

Factorization of weakly continuous differentiable mappings

Given real Banach spaces X and Y, let C _wbu¹(X, Y) be the space, introduced by R.M. Aron and J.B. Prolla, of C ¹ mappings from X into Y such that the mappings and their derivatives are weakly uniformly continuous on bounded sets. We show that f ∈ C _wbu¹(X, Y) if and only if f may be written in the form f = g ∘ S, where the intermediate space is normed, S is a precompact operator, and g is a Gateaux differentiable mapping with some additional properties.     

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

Transfer of complex structures and topological invariance of property of being holomorphic

The following question by V. I. Arnold is answered in affirmative. Let X, Y, and Z be three complex manifolds of equal dimension, let p: X → Y be a universal covering, and let g: Y → Z be a nondegenerate holornorphic mapping. Assume that the term Y in the chain X\xrightarrowpY\xrightarrowgZ X\xrightarrow{p}Y\xrightarrow{g}Z is “forgotten,” while the complex structures on X and Z are changed so that the mapping g ∘ p remains holomorphic. Can one recover the “forgotten” term Y? Bibliography: 2 titles.     

Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité

We prove Malgrange’s conjecture on the absence of confluence phenomena for integrable meromorphic connections. More precisely, if Y→X is a complex-analytic fibration by smooth curves, Z a hypersurface of Y finite over X, and ∇ an integrable meromorphic connection on Y with poles along Z, then the function which attaches to x∈X the sum of the irregularities of the fiber ∇_(x) at the points of Z _x is lower semicontinuous. The proof relies upon a study of the formal structure of integrable meromorphic connections in several variables.     

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

Oscillation for almost continuity

Let X be a topological space and (Y,d) be a metric space. If f: X → Y is a function then there is a function a _f : X → [0, ∞] such that f is almost continuous at x if and only if a _f (x) = 0. Some properties of this function are investigated. Supported by grant VEGA 2/6087/26 and APVT-51-006904.     

The structure of LC-continuous functions

A function f is LC-continuous if the inverse image of any open set is a locally closed set; i.e., an intersection of an open set and a closed set. The aim of this paper is to prove the following theorem: Let f: X→Y be an LC-continuous function onto a separable metric space Y. Then X can be covered by countably many subsets T _n ⊂X such that each restriction f∣T _n is continuous at all points of T _n .     

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan     

Extension ofUV n -valued mappings

A new method for extending upper semicontinuousUV ⁿ -valued mappings is introduced. Any upper semicontinuousUV ⁿ -valued mapping Ψ:A→Y of a closed subsetA of a separable metric spaceX into ann-connected, locallyn-connected complete metric spaceY satisfying the property of disjoint (n+1)-disks is proved to be extendable to an upper semicontinuousUV ⁿ -valued mapping Ψ′:X→Y such that Ψ′|a=Ψ. As an application, some results aboutn-soft mappings are obtained. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 351–363, September, 1999.     

A Note on Proper Maps of Locales

Let X and Y be completely regular locales. We show that the properness of a localic map f: X → Y can be characterized in terms of extension between compactifications.