Topological Aspects of Differential Chains
Identifieur interne : 002075 ( Istex/Corpus ); précédent : 002074; suivant : 002076Topological Aspects of Differential Chains
Auteurs : J. Harrison ; H. PughSource :
- Journal of Geometric Analysis [ 1050-6926 ] ; 2012-07-01.
English descriptors
Abstract
Abstract: In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus—the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U⊂M, but not generally in the strong dual topology of $\mathcal{B}'(U)$ .
Url:
DOI: 10.1007/s12220-010-9210-8
Links to Exploration step
ISTEX:9FD9918A5C517024BA97DEE81441DA8BE58555E5Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Topological Aspects of Differential Chains</title><author><name sortKey="Harrison, J" sort="Harrison, J" uniqKey="Harrison J" first="J." last="Harrison">J. Harrison</name><affiliation><mods:affiliation>Department of Mathematics, University of California, Berkeley, USA</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: harrison@math.berkeley.edu</mods:affiliation></affiliation></author><author><name sortKey="Pugh, H" sort="Pugh, H" uniqKey="Pugh H" first="H." last="Pugh">H. Pugh</name><affiliation><mods:affiliation>Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:9FD9918A5C517024BA97DEE81441DA8BE58555E5</idno><date when="2011" year="2011">2011</date><idno type="doi">10.1007/s12220-010-9210-8</idno><idno type="url">https://api.istex.fr/document/9FD9918A5C517024BA97DEE81441DA8BE58555E5/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">002075</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">002075</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Topological Aspects of Differential Chains</title><author><name sortKey="Harrison, J" sort="Harrison, J" uniqKey="Harrison J" first="J." last="Harrison">J. Harrison</name><affiliation><mods:affiliation>Department of Mathematics, University of California, Berkeley, USA</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: harrison@math.berkeley.edu</mods:affiliation></affiliation></author><author><name sortKey="Pugh, H" sort="Pugh, H" uniqKey="Pugh H" first="H." last="Pugh">H. Pugh</name><affiliation><mods:affiliation>Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Journal of Geometric Analysis</title><title level="j" type="abbrev">J Geom Anal</title><idno type="ISSN">1050-6926</idno><idno type="eISSN">1559-002X</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>New York</pubPlace><date type="published" when="2012-07-01">2012-07-01</date><biblScope unit="volume">22</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="685">685</biblScope><biblScope unit="page" to="690">690</biblScope></imprint><idno type="ISSN">1050-6926</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">1050-6926</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Bornological</term><term>Currents</term><term>Differential chains</term><term>Inductive limit</term><term>Mackey topology</term><term>Reflexive</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus—the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U⊂M, but not generally in the strong dual topology of $\mathcal{B}'(U)$ .</div></front></TEI><istex><corpusName>springer-journals</corpusName><author><json:item><name>J. Harrison</name><affiliations><json:string>Department of Mathematics, University of California, Berkeley, USA</json:string><json:string>E-mail: harrison@math.berkeley.edu</json:string></affiliations></json:item><json:item><name>H. Pugh</name><affiliations><json:string>Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>Differential chains</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Currents</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Mackey topology</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Bornological</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Reflexive</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Inductive limit</value></json:item></subject><articleId><json:string>9210</json:string><json:string>s12220-010-9210-8</json:string></articleId><arkIstex>ark:/67375/VQC-P92K6NXM-W</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus—the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U⊂M, but not generally in the strong dual topology of $\mathcal{B}'(U)$ .</abstract><qualityIndicators><refBibsNative>false</refBibsNative><abstractWordCount>147</abstractWordCount><abstractCharCount>1070</abstractCharCount><keywordCount>6</keywordCount><score>6.09</score><pdfWordCount>2326</pdfWordCount><pdfCharCount>10898</pdfCharCount><pdfVersion>1.4</pdfVersion><pdfPageCount>6</pdfPageCount><pdfPageSize>439.37 x 666.142 pts</pdfPageSize></qualityIndicators><title>Topological Aspects of Differential Chains</title><genre><json:string>research-article</json:string></genre><host><title>Journal of Geometric Analysis</title><language><json:string>unknown</json:string></language><publicationDate>2012</publicationDate><copyrightDate>2012</copyrightDate><issn><json:string>1050-6926</json:string></issn><eissn><json:string>1559-002X</json:string></eissn><journalId><json:string>12220</json:string></journalId><volume>22</volume><issue>3</issue><pages><first>685</first><last>690</last></pages><genre><json:string>journal</json:string></genre><subject><json:item><value>Dynamical Systems and Ergodic Theory</value></json:item><json:item><value>Abstract Harmonic Analysis</value></json:item><json:item><value>Fourier Analysis</value></json:item><json:item><value>Differential Geometry</value></json:item><json:item><value>Global Analysis and Analysis on Manifolds</value></json:item><json:item><value>Convex and Discrete Geometry</value></json:item></subject></host><ark><json:string>ark:/67375/VQC-P92K6NXM-W</json:string></ark><publicationDate>2012</publicationDate><copyrightDate>2011</copyrightDate><doi><json:string>10.1007/s12220-010-9210-8</json:string></doi><id>9FD9918A5C517024BA97DEE81441DA8BE58555E5</id><score>1</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/9FD9918A5C517024BA97DEE81441DA8BE58555E5/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/9FD9918A5C517024BA97DEE81441DA8BE58555E5/fulltext/zip</uri></json:item><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/9FD9918A5C517024BA97DEE81441DA8BE58555E5/fulltext/txt</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/9FD9918A5C517024BA97DEE81441DA8BE58555E5/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Topological Aspects of Differential Chains</title><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt></titleStmt><publicationStmt><authority>ISTEX</authority><publisher scheme="https://publisher-list.data.istex.fr">Springer-Verlag</publisher><pubPlace>New York</pubPlace><availability><licence><p>Mathematica Josephina, Inc., 2011</p></licence><p scheme="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</p></availability><date>2010-10-24</date></publicationStmt><notesStmt><note type="research-article" scheme="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</note><note type="journal" scheme="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Topological Aspects of Differential Chains</title><author xml:id="author-0000" corresp="yes"><persName><forename type="first">J.</forename><surname>Harrison</surname></persName><email>harrison@math.berkeley.edu</email><affiliation>Department of Mathematics, University of California, Berkeley, USA</affiliation></author><author xml:id="author-0001"><persName><forename type="first">H.</forename><surname>Pugh</surname></persName><affiliation>Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK</affiliation></author><idno type="istex">9FD9918A5C517024BA97DEE81441DA8BE58555E5</idno><idno type="ark">ark:/67375/VQC-P92K6NXM-W</idno><idno type="DOI">10.1007/s12220-010-9210-8</idno><idno type="article-id">9210</idno><idno type="article-id">s12220-010-9210-8</idno></analytic><monogr><title level="j">Journal of Geometric Analysis</title><title level="j" type="abbrev">J Geom Anal</title><idno type="pISSN">1050-6926</idno><idno type="eISSN">1559-002X</idno><idno type="journal-ID">true</idno><idno type="issue-article-count">13</idno><idno type="volume-issue-count">4</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>New York</pubPlace><date type="published" when="2012-07-01"></date><biblScope unit="volume">22</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="685">685</biblScope><biblScope unit="page" to="690">690</biblScope></imprint></monogr></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2010-10-24</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Abstract: In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus—the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U⊂M, but not generally in the strong dual topology of $\mathcal{B}'(U)$ .</p></abstract><textClass xml:lang="en"><keywords scheme="keyword"><list><head>Keywords</head><item><term>Differential chains</term></item><item><term>Currents</term></item><item><term>Mackey topology</term></item><item><term>Bornological</term></item><item><term>Reflexive</term></item><item><term>Inductive limit</term></item></list></keywords></textClass><textClass><classCode scheme="Mathematics Subject Classification (2000)">57N17</classCode><classCode scheme="Mathematics Subject Classification (2000)">58B10</classCode><classCode scheme="Mathematics Subject Classification (2000)">58A07</classCode><classCode scheme="Mathematics Subject Classification (2000)">58A10</classCode><classCode scheme="Mathematics Subject Classification (2000)">46E99</classCode></textClass><textClass><keywords scheme="Journal Subject"><list><head>Mathematics</head><item><term>Dynamical Systems and Ergodic Theory</term></item><item><term>Abstract Harmonic Analysis</term></item><item><term>Fourier Analysis</term></item><item><term>Differential Geometry</term></item><item><term>Global Analysis and Analysis on Manifolds</term></item><item><term>Convex and Discrete Geometry</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2010-10-24">Created</change><change when="2012-07-01">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-12-2">References added</change></revisionDesc></teiHeader></istex:fulltextTEI></fulltext><metadata><istex:metadataXml wicri:clean="corpus springer-journals not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Springer-Verlag</PublisherName><PublisherLocation>New York</PublisherLocation></PublisherInfo><Journal OutputMedium="All"><JournalInfo JournalProductType="NonStandardArchiveJournal" NumberingStyle="ContentOnly"><JournalID>12220</JournalID><JournalPrintISSN>1050-6926</JournalPrintISSN><JournalElectronicISSN>1559-002X</JournalElectronicISSN><JournalTitle>Journal of Geometric Analysis</JournalTitle><JournalAbbreviatedTitle>J Geom Anal</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Mathematics</JournalSubject><JournalSubject Type="Secondary">Dynamical Systems and Ergodic Theory</JournalSubject><JournalSubject Type="Secondary">Abstract Harmonic Analysis</JournalSubject><JournalSubject Type="Secondary">Fourier Analysis</JournalSubject><JournalSubject Type="Secondary">Differential Geometry</JournalSubject><JournalSubject Type="Secondary">Global Analysis and Analysis on Manifolds</JournalSubject><JournalSubject Type="Secondary">Convex and Discrete Geometry</JournalSubject></JournalSubjectGroup></JournalInfo><Volume OutputMedium="All"><VolumeInfo TocLevels="0" VolumeType="Regular"><VolumeIDStart>22</VolumeIDStart><VolumeIDEnd>22</VolumeIDEnd><VolumeIssueCount>4</VolumeIssueCount></VolumeInfo><Issue IssueType="Regular" OutputMedium="All"><IssueInfo IssueType="Regular" TocLevels="0"><IssueIDStart>3</IssueIDStart><IssueIDEnd>3</IssueIDEnd><IssueArticleCount>13</IssueArticleCount><IssueHistory><OnlineDate><Year>2012</Year><Month>6</Month><Day>6</Day></OnlineDate><PrintDate><Year>2012</Year><Month>6</Month><Day>5</Day></PrintDate><CoverDate><Year>2012</Year><Month>7</Month></CoverDate><PricelistYear>2012</PricelistYear></IssueHistory><IssueCopyright><CopyrightHolderName>Mathematica Josephina, Inc.</CopyrightHolderName><CopyrightYear>2012</CopyrightYear></IssueCopyright></IssueInfo><Article ID="s12220-010-9210-8" OutputMedium="All"><ArticleInfo ArticleType="OriginalPaper" ContainsESM="No" Language="En" NumberingStyle="ContentOnly" TocLevels="0"><ArticleID>9210</ArticleID><ArticleDOI>10.1007/s12220-010-9210-8</ArticleDOI><ArticleSequenceNumber>4</ArticleSequenceNumber><ArticleTitle Language="En">Topological Aspects of Differential Chains</ArticleTitle><ArticleFirstPage>685</ArticleFirstPage><ArticleLastPage>690</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2010</Year><Month>12</Month><Day>16</Day></RegistrationDate><Received><Year>2010</Year><Month>10</Month><Day>24</Day></Received><OnlineDate><Year>2011</Year><Month>1</Month><Day>14</Day></OnlineDate></ArticleHistory><ArticleEditorialResponsibility>Steven G. Krantz</ArticleEditorialResponsibility><ArticleCopyright><CopyrightHolderName>Mathematica Josephina, Inc.</CopyrightHolderName><CopyrightYear>2011</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1" CorrespondingAffiliationID="Aff1"><AuthorName DisplayOrder="Western"><GivenName>J.</GivenName><FamilyName>Harrison</FamilyName></AuthorName><Contact><Email>harrison@math.berkeley.edu</Email></Contact></Author><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>H.</GivenName><FamilyName>Pugh</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>University of California</OrgName><OrgAddress><City>Berkeley</City><Country Code="US">USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>Department of Pure Mathematics and Mathematical Statistics</OrgDivision><OrgName>University of Cambridge</OrgName><OrgAddress><City>Cambridge</City><Country Code="GB">UK</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En" OutputMedium="All"><Heading>Abstract</Heading><Para>In this paper we investigate the topological properties of the space of differential chains <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="12220_2010_9210_Article_IEq1.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\,^{\prime}\mathcal{B}(U)$</EquationSource></InlineEquation> defined on an open subset <Emphasis Type="Italic">U</Emphasis> of a Riemannian manifold <Emphasis Type="Italic">M</Emphasis>. We show that <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="12220_2010_9210_Article_IEq2.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\,^{\prime}\mathcal {B}(U)$</EquationSource></InlineEquation> is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="12220_2010_9210_Article_IEq3.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\,^{\prime}\mathcal{B}(U) $</EquationSource></InlineEquation>, and prove that it is a separable ultrabornological (<Emphasis Type="Italic">DF</Emphasis>)-space.</Para><Para>Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus—the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents <InlineEquation ID="IEq4"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="12220_2010_9210_Article_IEq4.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{B}'(U)$</EquationSource></InlineEquation> or <InlineEquation ID="IEq5"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="12220_2010_9210_Article_IEq5.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{D}'(U)$</EquationSource></InlineEquation>. For example, chains supported in finitely many points are dense in <InlineEquation ID="IEq6"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="12220_2010_9210_Article_IEq6.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\,^{\prime}\mathcal{B}(U)$</EquationSource></InlineEquation> for all open <Emphasis Type="Italic">U</Emphasis>⊂<Emphasis Type="Italic">M</Emphasis>, but not generally in the strong dual topology of <InlineEquation ID="IEq7"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="12220_2010_9210_Article_IEq7.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{B}'(U)$</EquationSource></InlineEquation>.</Para></Abstract><KeywordGroup Language="En"><Heading>Keywords</Heading><Keyword>Differential chains</Keyword><Keyword>Currents</Keyword><Keyword>Mackey topology</Keyword><Keyword>Bornological</Keyword><Keyword>Reflexive</Keyword><Keyword>Inductive limit</Keyword></KeywordGroup><KeywordGroup Language="--"><Heading>Mathematics Subject Classification (2000)</Heading><Keyword>57N17</Keyword><Keyword>58B10</Keyword><Keyword>58A07</Keyword><Keyword>58A10</Keyword><Keyword>46E99</Keyword></KeywordGroup><ArticleNote Type="CommunicatedBy"><SimplePara>Communicated by Steven G. Krantz.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Topological Aspects of Differential Chains</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Topological Aspects of Differential Chains</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">J.</namePart><namePart type="family">Harrison</namePart><affiliation>Department of Mathematics, University of California, Berkeley, USA</affiliation><affiliation>E-mail: harrison@math.berkeley.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">H.</namePart><namePart type="family">Pugh</namePart><affiliation>Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer-Verlag</publisher><place><placeTerm type="text">New York</placeTerm></place><dateCreated encoding="w3cdtf">2010-10-24</dateCreated><dateIssued encoding="w3cdtf">2012-07-01</dateIssued><dateIssued encoding="w3cdtf">2011</dateIssued><copyrightDate encoding="w3cdtf">2011</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus—the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U⊂M, but not generally in the strong dual topology of $\mathcal{B}'(U)$ .</abstract><subject lang="en"><genre>Keywords</genre><topic>Differential chains</topic><topic>Currents</topic><topic>Mackey topology</topic><topic>Bornological</topic><topic>Reflexive</topic><topic>Inductive limit</topic></subject><classification displayLabel="Mathematics Subject Classification (2000)">57N17</classification><classification displayLabel="Mathematics Subject Classification (2000)">58B10</classification><classification displayLabel="Mathematics Subject Classification (2000)">58A07</classification><classification displayLabel="Mathematics Subject Classification (2000)">58A10</classification><classification displayLabel="Mathematics Subject Classification (2000)">46E99</classification><relatedItem type="host"><titleInfo><title>Journal of Geometric Analysis</title></titleInfo><titleInfo type="abbreviated"><title>J Geom Anal</title></titleInfo><genre type="journal" displayLabel="Non Standard Archive Journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>Springer</publisher><dateIssued encoding="w3cdtf">2012-06-06</dateIssued><copyrightDate encoding="w3cdtf">2012</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Dynamical Systems and Ergodic Theory</topic><topic>Abstract Harmonic Analysis</topic><topic>Fourier Analysis</topic><topic>Differential Geometry</topic><topic>Global Analysis and Analysis on Manifolds</topic><topic>Convex and Discrete Geometry</topic></subject><identifier type="ISSN">1050-6926</identifier><identifier type="eISSN">1559-002X</identifier><identifier type="JournalID">12220</identifier><identifier type="IssueArticleCount">13</identifier><identifier type="VolumeIssueCount">4</identifier><part><date>2012</date><detail type="volume"><number>22</number><caption>vol.</caption></detail><detail type="issue"><number>3</number><caption>no.</caption></detail><extent unit="pages"><start>685</start><end>690</end></extent></part><recordInfo><recordOrigin>Mathematica Josephina, Inc., 2012</recordOrigin></recordInfo></relatedItem><identifier type="istex">9FD9918A5C517024BA97DEE81441DA8BE58555E5</identifier><identifier type="ark">ark:/67375/VQC-P92K6NXM-W</identifier><identifier type="DOI">10.1007/s12220-010-9210-8</identifier><identifier type="ArticleID">9210</identifier><identifier type="ArticleID">s12220-010-9210-8</identifier><accessCondition type="use and reproduction" contentType="copyright">Mathematica Josephina, Inc., 2011</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</recordContentSource><recordOrigin>Mathematica Josephina, Inc., 2011</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/9FD9918A5C517024BA97DEE81441DA8BE58555E5/metadata/json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Mathematiques/explor/BourbakiV1/Data/Istex/Corpus
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 002075 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Istex/Corpus/biblio.hfd -nk 002075 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Wicri/Mathematiques
|area= BourbakiV1
|flux= Istex
|étape= Corpus
|type= RBID
|clé= ISTEX:9FD9918A5C517024BA97DEE81441DA8BE58555E5
|texte= Topological Aspects of Differential Chains
}}
| This area was generated with Dilib version V0.6.33. |  |