Continuity properties of measurable group cohomology

Tim Austin, Calvin C. Moore

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)885-937
Number of pages53
JournalMathematische Annalen
Volume356
Issue number3
DOIs
StatePublished - Jul 2013

Fingerprint

Group Cohomology
Cohomology Group
Cohomology of Groups
Module
Cocycle
Sheaves
Quotient topology
Direct Limit
Inverse Limit
Classifying Space
Target
Locally Compact Group
Simplification
Vanish
Analogue
Range of data

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Continuity properties of measurable group cohomology. / Austin, Tim; Moore, Calvin C.

In: Mathematische Annalen, Vol. 356, No. 3, 07.2013, p. 885-937.

Research output: Contribution to journalArticle

Austin, Tim ; Moore, Calvin C. / Continuity properties of measurable group cohomology. In: Mathematische Annalen. 2013 ; Vol. 356, No. 3. pp. 885-937.
@article{bb3ae7dc27f54f8696c77564f8a853cd,
title = "Continuity properties of measurable group cohomology",
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 Fr{\'e}chet 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 Fr{\'e}chet, 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.",
author = "Tim Austin and Moore, {Calvin C.}",
year = "2013",
month = "7",
doi = "10.1007/s00208-012-0868-z",
language = "English (US)",
volume = "356",
pages = "885--937",
journal = "Mathematische Annalen",
issn = "0025-5831",
publisher = "Springer New York",
number = "3",

}

TY - JOUR

T1 - Continuity properties of measurable group cohomology

AU - Austin, Tim

AU - Moore, Calvin C.

PY - 2013/7

Y1 - 2013/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84878867934&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878867934&partnerID=8YFLogxK

U2 - 10.1007/s00208-012-0868-z

DO - 10.1007/s00208-012-0868-z

M3 - Article

VL - 356

SP - 885

EP - 937

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 3

ER -