The exponential law for partial, local and proper maps and its application to otopy theory
type: Item
instance of: scholarly article
title: The exponential law for partial, local and proper maps and its application to otopy theory
zbMATH Open document ID: 1303.55003
DOI: 10.1142/S0219199714500059
author: Piotr Bartłomiejczyk
author: Piotr Nowak-Przygodzki
published in: Communications in Contemporary Mathematics
publication date: 2014-10-15T00:00:00Z
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 6356386
zbMATH Keywords: partial map
zbMATH Keywords: local map
zbMATH Keywords: exponential law
zbMATH Keywords: sub bases
zbMATH Keywords: compact-open topology
MaRDI profile type: Publication
cites work: A Compact-Open Topology on Partial Maps with Open Domain
cites work: Gradient otopies of gradient local maps
cites work: Proper gradient otopies
cites work: Vector fields and transfers
cites work: Classification of homotopy classes of equivariant gradient maps
cites work: The topology of spaces of rational functions
cites work: On products in homotopy groups
full work available at URL: s0219199714500059
OpenAlex ID: W2095631614
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review text: ``The main goal of [the] paper is to introduce a topology on the set of local maps from \(X \rightarrow Y\)'', denoted here by \(Loc \, ( Y^I)\), so that the paths on the space correspond to otopies. Local mapping spaces gave rise to homotopies between two local maps, by means of the exponential law: \(Z^{X^I}\) is homeomorphic to \(Z^{(X\times I)}\). The local map \({(X\times I ) \rightarrow} Y\) looks like it should be a homotopy, but the domains at 0 and 1 have no connection with each other. An otopy is a special kind of homotopy so that the domains of the functions on \(0 \times X\) are connected to the domains on \(0 \times X\) by a sequence of connected domains on \( t \times X\). This is ratified by the fact that the sum of the local degrees of a proper gradient map is the same at each end of the otopy if and only if the top and bottom gradient maps have the same sum of degrees.NEWLINENEWLINEFirst the authors define the set of \textit{partial maps}, \(Par(Y^X)\) by \(f:D_f \rightarrow Y\) where \(f:D_f \) is an open subset of \(X\). The exponential law for sets holds, and so also for spaces where \( Y\) is a topological space and \(Z\) is a locally compact Hausdorff space and \(X\) is Hausdorff and \(Y^X \) is the space of maps with the compact-open topology.NEWLINENEWLINENow \(\kappa: Par(Y^{(X)} ) \rightarrow (Y^+)^X : x \mapsto f(x) x\) if \(x \in D_f\) and \(x\mapsto +\), if \(x \notin D_f\) for any \(f\), and where \(Y^+\) is \(Y\) with an outside point +. Now \(\kappa\) is well-defined and a homeomorphism. And since \( Par(Y^{(X)} )\) is homeomorphic to a mapping space, the exponential law follows.NEWLINENEWLINEThe authors introduce a subbasis for any space X with a different topology \(\tau(X)\). They call it the \textit{compact-open topology} for \(\tau(X)\). ``For any compact set \(C\) in \(X\), let \(H(C)=\{ U\in \tau(X)\mid C \Subset U \}\). Then the collection \(\{H(C)=|C\Subset X\}\) forms a subbasis for a topology on \(X: U \rightarrow \).''NEWL…
reviewed by: Daniel H. Gottlieb
description: scientific article; zbMATH DE number 6356386