About: Pullback (cohomology)

An Entity of Type: Thing, from Named Graph: http://dbpedia.org, within Data Space: dbpedia.org

In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism: from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map. For example, if X, Y are manifolds, R the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms.

Property Value
dbo:abstract
  • In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism: from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map. For example, if X, Y are manifolds, R the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms. The homotopy invariance of cohomology states that if two maps f, g: X → Y are homotopic to each other, then they determine the same pullback: f* = g*. In contrast, a pushforward for de Rham cohomology for example is given by integration-along-fibers. (en)
dbo:wikiPageID
  • 50783621 (xsd:integer)
dbo:wikiPageLength
  • 2375 (xsd:nonNegativeInteger)
dbo:wikiPageRevisionID
  • 784979954 (xsd:integer)
dbo:wikiPageWikiLink
dbp:wikiPageUsesTemplate
dcterms:subject
rdfs:comment
  • In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism: from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map. For example, if X, Y are manifolds, R the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms. (en)
rdfs:label
  • Pullback (cohomology) (en)
owl:sameAs
prov:wasDerivedFrom
foaf:isPrimaryTopicOf
is dbo:wikiPageWikiLink of
is foaf:primaryTopic of
Powered by OpenLink Virtuoso    This material is Open Knowledge     W3C Semantic Web Technology     This material is Open Knowledge    Valid XHTML + RDFa
This content was extracted from Wikipedia and is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License