arXiv subjects

Jerzy Dydak

Publications and source records attributed to Jerzy Dydak.

34 records · Page 2Linked to original sources

arXiv2015

Preserving Coarse Properties

The aim of this paper is to investigate properties preserved and co-preserved by coarsely $n$-to-1 functions, in particular by the quotient maps $X\to X/\sim$ induced by a finite group $G$ acting by isometries on a metric space $X$. The coarse properties we are mainly interested in are related to asymptotic dimension and its generalizations: having finite asymptotic dimension, asymptotic Property C, straight finite decomposition complexity, countable asymptotic dimension, and metric sparsification property. We provide an alternative description of asymptotic Property C and we prove that the class of spaces with straight finite decomposition complexity coincides with the class of spaces of countable asymptotic dimension.

math.MG↗

arXiv2015

Inducing maps between Gromov boundaries

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the class of visual functions $f$ that do induce continuous maps $\tilde f$ between Gromov boundaries. Its subclass, the class of radial functions, induces Hoelder maps between Gromov boundaries. Conversely, every Hoelder map between Gromov boundaries of visual hyperbolic spaces induces a radial function. We study the relationship between large scale properties of f and small scale properties of $f$, especially related to the dimension theory. In particular, we prove a form of the dimension raising theorem. We give a natural example of a radial dimension raising map and we also give a general class of radial functions that raise asymptotic dimension.

math.MG↗

arXiv2015

Axiomatization of geometry employing group actions

The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a topological, not a metric concept. That leads quickly to the notion of connectedness without any need to dwell on the definition of topology. In our approach line segments must be connected. Lines and planes are unified via the concept of separation: lines are separated into two components by each point, planes contain lines that separate them into two components as well. We add a subgroup of bijections preserving line segments and establishing unique isomorphism of basic geometrical sets, and the axiomatic structure is complete. Of fundamental importance is the Fixed Point Theorem that allows for creation of the concepts of length and congruency of line segments. The resulting structure is much more in sync with modern science than other axiomatic approaches to planar geometry. For instance, it leads naturally to the Erlangen Program in geometry. Our Conditions of Homogeneity and Rigidity have two interpretations. In physics, they correspond to the basic tenet that independent observers should arrive at the same measurement and are related to boosts in special relativity. In geometry, they mean uniqueness of congruence for certain geometrical figures. Another thread of the paper is the introduction of boundary at infinity, an important concept of modern mathematics, and linking of Pasch Axiom to endowing boundaries at infinity with a natural relation of betweenness. That way spherical geometry can be viewed as geometry of boundaries at infinity.

math.MG↗

arXiv2015

Coarse amenability and discreteness

This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of G.Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of G.Yu and straight finite decomposition complexity of Dranishnikov-Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

math.MG↗

arXiv2013

Partitions of unity and coverings

The aim of this paper is to prove all well-known metrization theorems using partitions of unity. To accomplish this, we first discuss sufficient and necessary conditions for existence of $\mathcal{U}$-small partitions of unity (partitions of unity subordinated to an open cover $\mathcal{U}$ of a topological space $X$).

math.GN↗

arXiv2013

Extensions of maps to M(Z_m,1)

We show that a Moore space M(Z_m,1) is an absolute extensor for finite dimensional metrizable spaces of cohomological dimension dim_{Z_m} \leq 1.

math.GT↗

arXiv2013

An approach to basic set theory and logic

The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows for quick proofs of basic set-theoretic identities and logical tautologies, so it is also a good tool to aid one's memory. I do not assume any exposure to euclidean geometry via axioms. Only an experience with transforming algebraic identities is required. The idea is to get students to do proofs right from the get-go. In particular, I avoid entangling students in nuances of logic early on. Basic facts of logic are derived from set theory, not the other way around.

math.LO↗

arXiv2013

Overlays and group actions

Overlays were introduced by R. H. Fox [6] as a subclass of covering maps. We offer a different view of overlays: it resembles the definition of paracompact spaces via star refinements of open covers. One introduces covering structures for covering maps and $p:X\to Y$ is an overlay if it has a covering structure that has a star refinement. We prove two characterizations of overlays: one using existence and uniqueness of lifts of discrete chains, the second as maps inducing simplicial coverings of nerves of certain covers. We use those characterizations to improve results of Eda-Matijevi\' c concerning topological group structures on domains of overlays whose range is a compact topological group. In case of surjective maps $p:X\to Y$ between connected metrizable spaces we characterize overlays as local isometries: $p:X\to Y$ is an overlay if and only if one can metrize $X$ and $Y$ in such a way that $p|B(x,1):B(x,1)\to B(p(x),1)$ is an isometry for each $x\in X$.

math.AT↗

arXiv2012

Anomalous coverings

We give examples of anomalous two-fold coverings $p:E\to B$ of connected spaces: a. one where $B$ is simply connected, b. the other of path-connected spaces that has an Evil Twin; a non equivalent covering $q:E\to B$ with the same image of the fundamental group.

math.AT↗

arXiv2011

Coverings and fundamental groups: a new approach

The traditional approach of defining the fundamental group first and then constructing universal coverings works well only for the class of Poincar\' e spaces. For general spaces there were several attempts to define generalized coverings (see \cite{BP3}, \cite{BDLM}, and \cite{FisZas}), yet there is no general theory so far that covers all path connected spaces. In this paper we plan to remedy that by changing the order of things: we define the universal covering first and its group of deck transformations is the new fundamental group of the base space. The basic idea is that a non-trivial loop ought to be detected by a covering (not by extension over the unit disk): a loop is non-trivial if there is a covering such that some lift of the loop is a non-loop. So it remains to define coverings: the most natural class is the class of maps that have unique disk lifting property. To make the theory work one needs to add the assumption that path components of pre-images of open sets form a basis of the total space.

math.GN↗

arXiv2007

Maps to the projective plane

We prove the projective plane $\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by A.N.Dranishnikov) in extension theory. One of the main tools is the computation of the fundamental group of the function space $\Map(\rp^n,\rp^{n+1})$ (based at inclusion) as being isomorphic to either $\Z_4$ or $\Z_2\oplus\Z_2$ for $n\ge 1$. Double surgery and the above fact yield the proof.

math.GT↗

arXiv2005

Extensions of maps to the projective plane

It is proved that for a 3-dimensional compact metrizable space X the infinite real projective space is an absolute extensor of X if and only if the real projective plane is an absolute extensor of X.

math.GT↗

arXiv2004

Algebra of dimension theory

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of those equivalence classes: \linebreak 1. For pointed CW complexes $K$ and $L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). \linebreak 2. For pointed compact spaces $X$ and $Y$, $\dim(\cal H^{-\ast}(X))=\dim(\cal H^{-\ast}(Y))$ if and only if $X$ and $Y$ are of the same dimension type (i.e., $\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$). Dranishnikov's version of Hurewicz Theorem in extension theory becomes $\dim(π_\ast(K))=\dim(H_\ast(K))$ for all simply connected $K$. The concept of cohomological dimension $\dim_A(X)$ of a pointed compact space $X$ with respect to a graded group $A$ is introduced. It turns out $\dim_A(X)\leq 0$ iff $\dim_{A(n)}(X)\leq n$ for all $n\in\Z$. If $A$ and $B$ are two positive graded groups, then $\dim(A)=\dim(B)$ if and only if $\dim_A(X)=\dim_B(X)$ for all compact $X$.

math.AT↗

arXiv2004

Extension theory of infinite symmetric products

We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension $\ExD(X)$ was introduced by A.N.Dranishnikov \cite {D$_5$} in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products $SP(L)$. One of the main ideas of the paper is to treat $\ExD(X)\leq SP(L)$ as the fundamental concept of cohomological dimension theory instead of $\dim_G(X)\leq n$. In a subsequent paper \cite{Dy$_6$} we show how properties of infinite symmetric products lead naturally to a calculus of graded groups which implies most of classical results of the cohomological dimension. The basic notion in \cite{Dy$_6$} is that of homological dimension of a graded group which allows for simultanous treatment of cohomological dimension of compacta and extension properties of CW complexes. We introduce cohomology of $X$ with respect to $L$ (defined as homotopy groups of the function space $SP(L)^X$). As an application of our results we characterize all countable groups $G$ so that the Moore space $M(G,n)$ is of the same extension type as the Eilenberg-MacLane space $K(G,n)$. Another application is characterization of infinite symmetric products of the same extension type as a compact (or finite-dimensional and countable) CW complex.

math.AT↗

arXiv2003

Partitions of unity

The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing major results on paracompactness (Stone Theorem and Tamano Theorem), a result which serves as a connection to Ascoli Theorem. A new calculus of partitions of unity is introduced with applications to dimension theory and metric simplicial complexes. The geometric interpretation of this calculus is the barycentric subdivision of simplicial complexes. Also, joins of partitions of unity are often used; they are an algebraic version of joins of simplicial complexes.

math.GN↗

arXiv2002

Extension dimension for paracompact spaces

We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: \proclaim{Theorem} Suppose X is a paracompact space. There is a CW complex K such that {a.} K is an absolute extensor of X up to homotopy, {b.} If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy. proclaim The proof is based on the following simple result (see 1.6). \proclaim{Theorem} Suppose X be a paracompact space and $f:A\to Y$ is a map from a closed subset A of X to a space Y. f extends over X if Y is the union of a family $\{Y_s\}_{s\in S}$ of its subspaces with the following properties: {a.} Each $Y_s$ is an absolute extensor of X, {b.} For any two elements s and t of S there is $u\in S$ such that $Y_s\cup Y_t\subset Y_u$, {c.} $A=\bigcup\limits_{s\in S} \int_A(f^{-1}(Y_s))$. proclaim That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.

math.GN↗