Functors of Complexes
Identifieur interne : 000B44 ( Istex/Corpus ); précédent : 000B43; suivant : 000B45Functors of Complexes
Auteurs : Albrecht DoldSource :
- Classics in Mathematics [ 1431-0821 ] ; 1995.
Abstract
Abstract: If T: ∂AG→∂AG is a functor from complexes to complexes then X↦TSX provides a generalization of the singular complex SX which may yield new useful topological invariants. We study this question (§§ 2–7), at least if T is the (dimension-wise) prolongation of an additive functor t: AG→AG. We find that for every abelian group G there is, essentially, one covariant and one contravariant t such that tℤ=G. The resulting groups HTSX are the homology respectively cohomology groups of X with coefficients in G. The functors t are also useful in studying product spaces; these questions are discussed in §§8–12.
Url:
DOI: 10.1007/978-3-642-67821-9_6
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We study this question (§§ 2–7), at least if T is the (dimension-wise) prolongation of an additive functor t: AG→AG. We find that for every abelian group G there is, essentially, one covariant and one contravariant t such that tℤ=G. The resulting groups HTSX are the homology respectively cohomology groups of X with coefficients in G. 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VI</ChapterNumber><ChapterDOI>10.1007/978-3-642-67821-9_6</ChapterDOI><ChapterSequenceNumber>6</ChapterSequenceNumber><ChapterTitle Language="En">Functors of Complexes</ChapterTitle><ChapterFirstPage>123</ChapterFirstPage><ChapterLastPage>185</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>1995</CopyrightYear></ChapterCopyright><ChapterHistory><RegistrationDate><Year>2012</Year><Month>4</Month><Day>20</Day></RegistrationDate></ChapterHistory><ChapterGrants 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></ChapterGrants><ChapterContext><SeriesID>3242</SeriesID><BookID>978-3-642-67821-9</BookID><BookTitle>Lectures on Algebraic Topology</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>Albrecht</GivenName><FamilyName>Dold</FamilyName></AuthorName></Author><Affiliation ID="Aff2"><OrgDivision>Mathematisches Institut</OrgDivision><OrgName>Universität Heidelberg</OrgName><OrgAddress><Street>Im Neuenheimer Feld 288</Street><Postcode>69120</Postcode><City>Heidelberg</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>If <Emphasis Type="Italic">T</Emphasis>: ∂<Emphasis FontCategory="NonProportional">AG</Emphasis>→∂<Emphasis FontCategory="NonProportional">AG</Emphasis> is a functor from complexes to complexes then <Emphasis Type="Italic">X</Emphasis>↦<Emphasis Type="Italic">TSX</Emphasis> provides a generalization of the singular complex <Emphasis Type="Italic">SX</Emphasis> which may yield new useful topological invariants. We study this question (§§ 2–7), at least if <Emphasis Type="Italic">T</Emphasis> is the (dimension-wise) prolongation of an additive functor <Emphasis Type="Italic">t</Emphasis>: <Emphasis FontCategory="NonProportional">AG</Emphasis>→<Emphasis FontCategory="NonProportional">AG</Emphasis>. We find that for every abelian group <Emphasis Type="Italic">G</Emphasis> there is, essentially, one covariant and one contravariant <Emphasis Type="Italic">t</Emphasis> such that tℤ=<Emphasis Type="Italic">G</Emphasis>. The resulting groups <Emphasis Type="Italic">HTSX</Emphasis> are the <Emphasis Type="Italic">homology respectively cohomology groups of X with coefficients in G</Emphasis>. The functors <Emphasis Type="Italic">t</Emphasis> are also useful in studying product spaces; these questions are discussed in §§8–12.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Functors of Complexes</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Functors of Complexes</title></titleInfo><name type="personal"><namePart type="given">Albrecht</namePart><namePart type="family">Dold</namePart><affiliation>Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120, Heidelberg, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="research-article" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">1995</dateIssued><dateIssued encoding="w3cdtf">1995</dateIssued><copyrightDate encoding="w3cdtf">1995</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: If T: ∂AG→∂AG is a functor from complexes to complexes then X↦TSX provides a generalization of the singular complex SX which may yield new useful topological invariants. We study this question (§§ 2–7), at least if T is the (dimension-wise) prolongation of an additive functor t: AG→AG. We find that for every abelian group G there is, essentially, one covariant and one contravariant t such that tℤ=G. The resulting groups HTSX are the homology respectively cohomology groups of X with coefficients in G. 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