About: L² cohomology

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In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form.

Attributes	Values
rdf:type	work yago:WikicatCohomologyTheories yago:Abstraction100002137 yago:Cognition100023271 yago:Explanation105793000 yago:HigherCognitiveProcess105770664 yago:Process105701363 yago:PsychologicalFeature100023100 yago:Theory105989479 yago:Thinking105770926
rdfs:label	L² cohomology (en) L²-kohomologi (sv)
rdfs:comment	In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form. (en) Inom matematiken är L2–kohomologi en kohomologiteori för okompakta mångfalder M med Riemannmetrik. Den definieras på samma sätt som de Rhamkohomologi förutom att man använder kvadratiskt integrerbara differentialformer. Beteckningen av kvadratiskt integrerbar kan användas eftersom ur metriken över M uppstår en norm över differentialformer och en . L2-kohomologi studerades oberoende av (1978) och (1979). Den är nära relaterad till . (sv)
dcterms:subject	Differential geometry Cohomology theories Differential topology
Wikipage page ID	10465223 (xsd:integer)
Wikipage revision ID	1094074841 (xsd:integer)
Link from a Wikipage to another Wikipage	De Rham cohomology Annals of Mathematics Volume form Inventiones Mathematicae Differential geometry Mathematics Vertical bar Cohomology theories Steven Zucker Compositio Mathematica Mark Stern Differential topology Frances Kirwan Dirichlet form Eduard Looijenga Journal of Differential Geometry Mark Goresky Riemannian manifold Riemannian metric Henri Cartan Isomorphic Jeff Cheeger Baily–Borel compactification Hermitian variety Differential form Manifold Square-integrable Smooth manifold Metric (mathematics) Intersection cohomology Middle perversity Dirichlet principle Cohomology theory Locally symmetric variety
Link from a Wikipage to an external page	http://www.math.ias.edu/~goresky/pdf/zucker.pdf
sameAs	L² cohomology L² cohomology L² cohomology L² cohomology
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Cite_book dbt:Cite_journal dbt:Refbegin dbt:Refend dbt:Isbn dbt:Differential-geometry-stub dbt:Topology-stub
first	B. Brent (en)
id	B/b130010 (en)
last	Gordon (en)
title	Baily–Borel compactification (en)
has abstract	In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form. L2 cohomology, which grew in part out of L2 d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker (1978) and Jeff Cheeger (1979). It is closely related to intersection cohomology; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology. Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L2 cohomology is isomorphic to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by Eduard Looijenga (1988) and by Leslie Saper and Mark Stern (1990). (en) Inom matematiken är L2–kohomologi en kohomologiteori för okompakta mångfalder M med Riemannmetrik. Den definieras på samma sätt som de Rhamkohomologi förutom att man använder kvadratiskt integrerbara differentialformer. Beteckningen av kvadratiskt integrerbar kan användas eftersom ur metriken över M uppstår en norm över differentialformer och en . L2-kohomologi studerades oberoende av (1978) och (1979). Den är nära relaterad till . Ett resultat inom L2–kohomologi är Zuckers förmodan, som säger att för en är L2–kohomologin isomorfisk till snittkohomologin av dess Baily–Borelkompaktifiering (Zucker 1982). Detta bevisades på olika sätt av Looijenga (1988) och Saper och Stern (1990). (sv)
gold:hypernym	Mythology
prov:wasDerivedFrom	wikipedia-en:L²_cohomology?oldid=1094074841&ns=0
page length (characters) of wiki page	5155 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:L²_cohomology
is Link from a Wikipage to another Wikipage of	L2 L2 cohomology Zucker conjecture Perverse sheaf Jeff Cheeger Baily–Borel compactification Wolfgang Lück L2-Betti number L2-cohomology
is Wikipage redirect of	L2 cohomology Zucker conjecture L2-Betti number L2-cohomology
is foaf:primaryTopic of	wikipedia-en:L²_cohomology

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