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1. CMB 2013 (vol 57 pp. 364)

Li, Lei; Wang, Ya-Shu
How Lipschitz Functions Characterize the Underlying Metric Spaces
Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that both $X,Y$ are realcompact, or both $E,F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z(f)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if \[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\] or \[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\] respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim E =\dim F\lt +\infty$, is a weighted composition operator $(Tf)(y)=J_y(f(\tau(y)))$. We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.

Keywords:(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps
Categories:46E40, 54D60, 46E15

2. CMB 2011 (vol 56 pp. 55)

Bouziad, A.
Cliquishness and Quasicontinuity of Two-Variable Maps
We study the existence of continuity points for mappings $f\colon X\times Y\to Z$ whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are fragmentable and $y$-sections $X\ni x\to f(x,y)\in Z$ are quasicontinuous, where $X$ is a Baire space and $Z$ is a metric space. For the factor $Y$, we consider two infinite ``point-picking'' games $G_1(y)$ and $G_2(y)$ defined respectively for each $y\in Y$ as follows: in the $n$-th inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$. Then Player II picks a point $y_n\in D_n$; II wins if $y$ is in the closure of ${\{y_n:n\in\mathbb N\}}$, otherwise I wins. It is shown that (i) $f$ is cliquish if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $ f$ is quasicontinuous if the $x$-sections of $f$ are continuous and the set of $y\in Y$ such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially a result of Debs and item (ii) indicates that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of ``small'' compact spaces.

Keywords:cliquishness, fragmentability, joint continuity, point-picking game, quasicontinuity, separate continuity, two variable maps
Categories:54C05, 54C08, 54B10, 91A05

3. CMB 2011 (vol 55 pp. 523)

Iwase, Norio; Mimura, Mamoru; Oda, Nobuyuki; Yoon, Yeon Soo
The Milnor-Stasheff Filtration on Spaces and Generalized Cyclic Maps
The concept of $C_{k}$-spaces is introduced, situated at an intermediate stage between $H$-spaces and $T$-spaces. The $C_{k}$-space corresponds to the $k$-th Milnor-Stasheff filtration on spaces. It is proved that a space $X$ is a $C_{k}$-space if and only if the Gottlieb set $G(Z,X)=[Z,X]$ for any space $Z$ with ${\rm cat}\, Z\le k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\Sigma B,X)=[\Sigma B,X]$ for any space $B$. Some results on the $C_{k}$-space are generalized to the $C_{k}^{f}$-space for a map $f\colon A \to X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of $C_{k}$-spaces, and non-$C_{k}$-spaces.

Keywords:Gottlieb sets for maps, L-S category, T-spaces
Categories:55P45, 55P35

4. CMB 2011 (vol 54 pp. 607)

Camargo, Javier
Lightness of Induced Maps and Homeomorphisms
An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and $2^{f}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive integer $n$, we study when the lightness of the induced map $2^{f}$ or $C_n(f)$ implies that $f$ is a homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.

Keywords:light maps, induced maps, continua, hyperspaces
Categories:54B20, 54E40

5. CMB 2010 (vol 53 pp. 466)

Dubarbie, Luis
Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions
In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.

Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuity
Categories:47B38, 46E15, 46E40, 46H40, 47B33

6. CMB 2009 (vol 52 pp. 18)

Chinea, Domingo
Harmonicity of Holomorphic Maps Between Almost Hermitian Manifolds
In this paper we study holomorphic maps between almost Hermitian manifolds. We obtain a new criterion for the harmonicity of such holomorphic maps, and we deduce some applications to horizontally conformal holomorphic submersions.

Keywords:almost Hermitian manifolds, harmonic maps, harmonic morphism
Categories:53C15, 58E20

7. CMB 2008 (vol 51 pp. 448)

Sasahara, Toru
Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms
Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

Keywords:biharmonic maps, Sasakian manifolds, Legendrian submanifolds
Categories:53C42, 53C40

8. CMB 2005 (vol 48 pp. 614)

Tuncali, H. Murat; Valov, Vesko
On Finite-to-One Maps
Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that for every positive integer $p$ with $ p\leq m+k+1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$ with the source limitation topology such that each fiber of $f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most $\max\{k+m-p+2,1\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let $f\colon X\to Y$ be a $k$-dimensional map between compact metric spaces with $\dim Y\leq m$. Then: \begin{inparaenum}[\rm(1)] \item there exists a map $h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to Y\times\uin^{m+2k}$ is 2-to-one provided $k\geq 1$; \item there exists a map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to Y\times\uin^{m+k+1}$ is $(k+1)$-to-one. \end{inparaenum}

Keywords:finite-to-one maps, dimension, set-valued maps
Categories:54F45, 55M10, 54C65

9. CMB 2002 (vol 45 pp. 97)

Haas, Andrew
Invariant Measures and Natural Extensions
We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real M\"obius transformations. Included are the maps that are exactly $n$-to-$1$, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.

Keywords:Continued fractions, interval maps, invariant measures
Categories:11J70, 58F11, 58F03

10. CMB 2001 (vol 44 pp. 266)

Cencelj, M.; Dranishnikov, A. N.
Extension of Maps to Nilpotent Spaces
We show that every compactum has cohomological dimension $1$ with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension $1$ with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely generated homotopy groups over all of $X$.

Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space
Categories:55M10, 55S36, 54C20, 54F45

11. CMB 2000 (vol 43 pp. 294)

Bracci, Filippo
Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity
We identify a class of domains of $\C^n$ in which any two commuting holomorphic self-maps have a common fixed point.

Keywords:Holomorphic self-maps, commuting functions, fixed points, Wolff point, Julia's Lemma
Categories:32A10, 32A40, 32H15, 32A30

12. CMB 1999 (vol 42 pp. 274)

Dădărlat, Marius; Eilers, Søren
The Bockstein Map is Necessary
We construct two non-isomorphic nuclear, stably finite, real rank zero $C^\ast$-algebras $E$ and $E'$ for which there is an isomorphism of ordered groups $\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to \bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible with all the coefficient transformations. The $C^\ast$-algebras $E$ and $E'$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras with similar properties.

Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$-algebras, real rank zero, purely infinite, classification
Categories:46L35, 46L80, 19K14

13. CMB 1998 (vol 41 pp. 442)

Chamberland, Marc; Meisters, Gary
A Mountain Pass to the Jacobian Conjecture.
This paper presents an approach to injectivity theorems via the Mountain Pass Lemma and raises an open question. The main result of this paper (Theorem~1.1) is proved by means of the Mountain Pass Lemma and states that if the eigenvalues of $F' (\x)F' (\x)^{T}$ are uniformly bounded away from zero for $\x \in \hbox{\Bbbvii R}^{n}$, where $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$ is a class $\cC^{1}$ map, then $F$ is injective. This was discovered in a joint attempt by the authors to prove a stronger result conjectured by the first author: Namely, that a sufficient condition for injectivity of class $\cC^{1}$ maps $F$ of $\hbox{\Bbbvii R}^n$ into itself is that all the eigenvalues of $F'(\x)$ are bounded away from zero on $\hbox{\Bbbvii R}^n$. This is stated as Conjecture~2.1. If true, it would imply (via {\it Reduction-of-Degree}) {\it injectivity of polynomial maps} $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$ {\it satisfying the hypothesis}, $\det F'(\x) \equiv 1$, of the celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The paper ends with several examples to illustrate a variety of cases and known counterexamples to some natural questions.

Keywords:Injectivity, ${\cal C}^1$-maps, polynomial maps, Jacobian Conjecture, Mountain Pass
Categories:14A25, 14E09

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