Continuity properties of measurable group cohomology
Identifieur interne : 000200 ( Main/Exploration ); précédent : 000199; suivant : 000201Continuity properties of measurable group cohomology
Auteurs : Tim Austin [États-Unis] ; Calvin C. Moore [États-Unis]Source :
- Mathematische Annalen [ 0025-5831 ] ; 2013-07-01.
Abstract
Abstract: A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood. This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to ‘regularize’ measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of consequences are then derived from this argument. First, we prove that for target modules that are Fréchet spaces, the cohomology groups agree with those defined using continuous cocycles, and hence they vanish in positive degrees when the acting group is compact. Using this, we then show that for Fréchet, discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups, and also under forming direct limits of discrete target modules. Lastly, these results together enable us to establish various circumstances under which the measurable-cochains cohomology groups coincide with others defined using sheaves on a semi-simplicial space associated to the underlying group, or sheaves on a classifying space for that group. We also prove in some cases that the natural quotient topologies on the measurable-cochains cohomology groups are Hausdorff.
Url:
DOI: 10.1007/s00208-012-0868-z
Affiliations:
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Moore</name><affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Mathematics, University of California, 94720, Berkeley, CA</wicri:regionArea><placeName><region type="state">Californie</region></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country></affiliation></author></analytic><monogr></monogr><series><title level="j">Mathematische Annalen</title><title level="j" type="abbrev">Math. Ann.</title><idno type="ISSN">0025-5831</idno><idno type="eISSN">1432-1807</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>Berlin/Heidelberg</pubPlace><date type="published" when="2013-07-01">2013-07-01</date><biblScope unit="volume">356</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="885">885</biblScope><biblScope unit="page" to="937">937</biblScope></imprint><idno type="ISSN">0025-5831</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0025-5831</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood. This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to ‘regularize’ measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of consequences are then derived from this argument. First, we prove that for target modules that are Fréchet spaces, the cohomology groups agree with those defined using continuous cocycles, and hence they vanish in positive degrees when the acting group is compact. Using this, we then show that for Fréchet, discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups, and also under forming direct limits of discrete target modules. Lastly, these results together enable us to establish various circumstances under which the measurable-cochains cohomology groups coincide with others defined using sheaves on a semi-simplicial space associated to the underlying group, or sheaves on a classifying space for that group. We also prove in some cases that the natural quotient topologies on the measurable-cochains cohomology groups are Hausdorff.</div></front></TEI><affiliations><list><country><li>États-Unis</li></country><region><li>Californie</li><li>État de New York</li></region></list><tree><country name="États-Unis"><region name="État de New York"><name sortKey="Austin, Tim" sort="Austin, Tim" uniqKey="Austin T" first="Tim" last="Austin">Tim Austin</name></region><name sortKey="Austin, Tim" sort="Austin, Tim" uniqKey="Austin T" first="Tim" last="Austin">Tim Austin</name><name sortKey="Moore, Calvin C" sort="Moore, Calvin C" uniqKey="Moore C" first="Calvin C." last="Moore">Calvin C. Moore</name><name sortKey="Moore, Calvin C" sort="Moore, Calvin C" uniqKey="Moore C" first="Calvin C." last="Moore">Calvin C. Moore</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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