On the algebraic foundations of bounded cohomology

126 Pages·2011·0.86 MB·English

by  Theo Bühler

MEMOIRS
of the
American Mathematical Society
Number 1006
On the Algebraic Foundations
of Bounded Cohomology
Theo Bu¨hler
November 2011 • Volume 214 • Number 1006 (second of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 1006
On the Algebraic Foundations
of Bounded Cohomology
Theo Bu¨hler
November2011 • Volume214 • Number1006(secondof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Bu¨hler,Theo,1978-
Onthealgebraicfoundationsofboundedcohomology/TheoBu¨hler.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 1006)
“November2011,volume214,number1006(secondof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-5311-5(alk. paper)
1. Derived categories (Mathematics). 2. Homology theory. 3. Functions of bounded varia-
tion. 4.Topologicalspaces. I.Title.
QA169.B84 2011
514(cid:2).23—dc23 2011030199
Memoirs of the American Mathematical Society
Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics.
Publisher Item Identifier. The Publisher Item Identifier (PII) appears as a footnote on
theAbstractpageofeacharticle. Thisalphanumericstringofcharactersuniquelyidentifieseach
articleandcanbeusedforfuturecataloguing,searching,andelectronicretrieval.
Subscription information. Beginning with the January 2010 issue, Memoirs is accessi-
ble from www.ams.org/journals. The 2011 subscription begins with volume 209 and consists of
six mailings, each containing one or more numbers. Subscription prices are as follows: for pa-
per delivery, US$741 list, US$592.80 institutional member; for electronic delivery, US$667 list,
US$533.60institutional member. Uponrequest, subscribers topaper delivery ofthis journalare
also entitled to receive electronic delivery. If ordering the paper version, subscribers outside the
United States and India must pay a postage surcharge of US$69; subscribers in India must pay
apostagesurchargeofUS$95. ExpediteddeliverytodestinationsinNorthAmericaUS$58;else-
whereUS$167. Subscriptionrenewalsaresubjecttolatefees. Seewww.ams.org/help-faqformore
journalsubscriptioninformation. Eachnumbermaybeorderedseparately;pleasespecifynumber
whenorderinganindividualnumber.
Back number information. Forbackissuesseewww.ams.org/bookstore.
Subscriptions and orders should be addressed to the American Mathematical Society, P.O.
Box 845904, Boston, MA 02284-5904USA. All orders must be accompanied by payment. Other
correspondenceshouldbeaddressedto201CharlesStreet,Providence,RI02904-2294USA.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
MemoirsoftheAmericanMathematicalSociety(ISSN0065-9266)ispublishedbimonthly(each
volume consisting usually of more than one number) by the American Mathematical Society at
201CharlesStreet,Providence,RI02904-2294USA.PeriodicalspostagepaidatProvidence,RI.
Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles
Street,Providence,RI02904-2294USA.
(cid:2)c 2011bytheAmericanMathematicalSociety. Allrightsreserved.
Copyrightofindividualarticlesmayreverttothepublicdomain28years
afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles.
(cid:2) (cid:2) (cid:2)
ThispublicationisindexedinScienceCitation IndexR,SciSearchR,ResearchAlertR,
(cid:2) (cid:2)
CompuMath Citation IndexR,Current ContentsR/Physical,Chemical& Earth Sciences.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 161514131211
Contents
Introduction and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. Reconstructing the Burger-Monod Theory . . . . . . . . . . . . . . . . viii
2. More Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
3. Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
4. Amenable Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
5. (cid:2)1-Homology and Bounded Cohomology for Spaces . . . . . . . . . . .xviii
6. The Canonical Semi-Norm . . . . . . . . . . . . . . . . . . . . . . . . . xx
Part 1. Triangulated Categories . . . . . . . . . . . . . . . . . . . . 1
Chapter I. Triangulated Categories . . . . . . . . . . . . . . . . . . . . . . . 3
1. Definition and Elementary Properties. . . . . . . . . . . . . . . . . . . 3
2. Triangle Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter II. The Derived Category of an Exact Category . . . . . . . . . . . 17
1. Verdier Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2. The Derived Category of an Exact Category . . . . . . . . . . . . . . . 24
3. Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Chapter III. Abstract Truncation: t-Structures and Hearts . . . . . . . . . . 31
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2. Abstract Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Part 2. Homological Algebra for Bounded Cohomology . . . . . . 45
Chapter IV. Categories of Banach Spaces . . . . . . . . . . . . . . . . . . . 47
1. The Bicomplete Category of Banach Spaces . . . . . . . . . . . . . . . 47
2. The Additive Category of Banach Spaces . . . . . . . . . . . . . . . . 54
Chapter V. Derived Categories of Banach G-Modules . . . . . . . . . . . . . 63
1. Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2. Some Remarks on Topological Groups . . . . . . . . . . . . . . . . . . 70
Part 3. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Appendix A. Mapping Cones, Homotopy Push-Outs, Mapping Cylinders . . 75
1. The Mapping Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Appendix B. Pull-Back of Exact Structures . . . . . . . . . . . . . . . . . . 79
Appendix C. Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . 81
1. Statement of the Result . . . . . . . . . . . . . . . . . . . . . . . . . . 81
iii
iv CONTENTS
2. Proof of Theorem 1.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Appendix D. Standard Borel G-Spaces are Regular . . . . . . . . . . . . . . 87
1. Representations Associated to a Group Action . . . . . . . . . . . . . 87
Appendix E. The Existence of Bruhat Functions . . . . . . . . . . . . . . . 91
1. Proper Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Abstract
Bounded cohomology for topological spaces was introduced by Gromov in the
late seventies, mainly to describe the simplicial volume invariant. It is an exotic
cohomologytheoryforspacesinthatitfailsexcisionandthuscannotberepresented
by spectra. Gromov’s basic vanishing result of bounded cohomology for simply
connectedspacesimpliesthatboundedcohomologyforspacesisaninvariantofthe
fundamental group. To prove this, one is led to introduce a cohomology theory for
groups and the present work is concerned with the latter, which has been studied
by Gromov, Brooks, Ivanov and Noskov to name but the most important initial
contributors. Generalizing these ideas from discrete groups to topological groups,
Burger and Monod have developed continuous bounded cohomology in the late
nineties.
It is a widespread opinion among experts that (continuous) bounded cohomo-
logy cannot be interpreted as a derived functor and that triangulated methods
break down. We prove that this is wrong.
We use the formalism of exact categories and their derived categories in order
to construct a classical derived functor on the category of Banach G-modules with
valuesinWaelbroeck’sabeliancategory. Thisgivesusanaxiomaticcharacterization
of this theory for free and it is a simple matter to reconstruct the classical semi-
normed cohomology spaces out of Waelbroeck’s category.
Weprovethatthederivedcategoriesofrightboundedandofleftboundedcom-
plexes of Banach G-modules are equivalent to the derived category of two abelian
categories (one for each boundedness condition), a consequence of the theory of
abstracttruncationandheartsoft-structures. Moreover,weprovethatthederived
categories of Banach G-modules can be constructed as the homotopy categories of
model structures on the categories of chain complexes of Banach G-modules thus
proving that the theory fits into yet another standard framework of homological
and homotopical algebra.
ReceivedbytheeditorFebruary8,2008.
ArticleelectronicallypublishedonMarch11,2011;S0065-9266(2011)00618-0.
2000MathematicsSubjectClassification. Primary18E30,Secondary18E10,18G60,46M18,
20J05,57T.
Key wordsand phrases. BoundedCohomology,DerivedCategories,DerivedFunctors.
Affiliation at time of publication: Departement Mathematik der ETH Zu¨rich,
R¨amistrasse101,CH-8092Zu¨rich;email: [email protected].
(cid:2)c2011 American Mathematical Society
v

Introduction and Main Results
This work is based on the remarkable thesis of N. Monod [Mon01] which laid
out the foundations for the theory of continuous bounded cohomology with a view
towardsapplicationsinrigiditytheory. Itisthedetailedpresentationofjointresults
with his advisor M. Burger. The power and the merits of the machinery are well-
known and the reader should consult the recent works of Burger, Monod and their
collaboratorsin order toget anidea of the state of the art. Since Monod’s thesis is
quite heavy-going and the present work is probably even worse in this respect, we
recommend the novice to (cid:2)1-homology and bounded cohomology to consult Lo¨h’s
beautiful thesis [L¨oh07], the declared—and happily also achieved—goal of which
is to provide a lightweight approach to the theory.
Weshallfocusonthealgebraicaspectsofthetheoryinordertoprovideagood
reasonforthecloseresemblanceofthefundamentalhomological resultsinbounded
cohomology to their classical algebraic counterparts. Burger and Monod point
out that the definitions of the central notions are inspired by relative homological
algebra—cf. also N. V. Ivanov [Iva85]. Our point is that they are nothing but
instances of the categorical notions up to a slight detail which stems from the
interest in semi-norms.
Keeping track of semi-norms in homology is certainly of utmost importance to
most applications up to date. Many technicalities are due to them and it seems
that they are nothing but a nuisance for theoretical considerations. A clear-cut
approach is probably best achieved by dropping the semi-norms and by recovering
them only after the cohomology theory has been developed in full.
The main upshot of the present work is as follows:
(1) Underlyingcontinuous bounded cohomology there is an exactcategory in
the sense of Quillen with enough injectives.
(2) Continuous bounded cohomology arises from the total derived functor of
the maximal invariant submodule functor.
(3) The heart of the canonical left t-structure on the derived category of
BanachspacesisWaelbroeck’sabeliancategoryofquotientBanachspaces.
Continuous bounded cohomology factors over this abelian category and
canthusbeviewedasaclassicalderivedfunctor—modulothefactthatits
domain of definition is not an abelian category. An ad hoc construction
gives back the semi-normed space-valued functor of Burger and Monod.
Continuousboundedcohomologyisafunctionalanalyticvariantofgroupcohomol-
ogy. Objects of study are Hausdorff topological groups and their strongly continu-
ous and isometric representations on Banach spaces. Since the category of Banach
spaces is not abelian, the classical language does not apply. However, the category
ofBanachG-modulescanbeequippedwithvariousexactstructuresinthesenseof
vii
viii INTRODUCTION AND MAIN RESULTS
Quillen, for which the theory of the derived category allows us to speak of derived
functors.
At the heart of the matter is Burger-Monod’s “functorial characterization”
which tells us that continuous bounded cohomology may be computed as if it were
the classical derived functor of the G-invariants. By the work of Burger-Monod we
know that cohomology in degree zero coincides with the G-invariants, cohomology
vanishes on injectives in higher degrees and that there is a natural long exact
sequence for every short exactsequence of G-modules. Thus, bounded cohomology
shouldbethederivedfunctoroftheG-invariantsandthemainpurposeofthiswork
is to make this precise and to show that bounded cohomology for groups is not as
exotic as is often claimed.
1. Reconstructing the Burger-Monod Theory
Our construction of bounded cohomology involves three essential choices and
one ad hoc construction. These choices are justified by the original work of Burger
and Monod, see [Mon01] and [BM02].
Fix as ground field k either the real or complex numbers. Let Ban be the
category of Banach spaces over k and bounded k-linear maps.
Let G be a Hausdorff topological group and define the category G−Ban of
Banach G-modules to be the strongly continuous and isometric representations
of G on Banach spaces over k together with the bounded linear G-equivariant
maps. This is an additive category which is finitely bicomplete and whose class
of kernel-cokernel pairs coincides with the short sequences of Banach G-modules
whose underlying sequence of k-vector spaces is short exact.
Notice that there is a functor (−)G :G−Ban→Ban associating to a Banach
G-module M the closed subspace MG of G-invariant vectors. It is right adjoint to
thetrivialmodulefunctorBan→G−BanwhicharisesfromconsideringaBanach
space as a Banach G-module with the trivial G-action.
The three choices are:
(i) On G−Ban choose the exact structure given by the class EG of short se-
rel
quences whose underlying sequence of Banach spaces is split exact. Burger
andMonodhaveessentially shownthat(G−Ban,EG )hasenoughinjectives.
rel
Therefore the derived category D+(G−Ban,EG ) of left bounded complexes
rel
is equivalent to the homotopy category K+(I (EG )) of left bounded com-
rel
plexes of EG -injective Banach G-modules. In particular D+(G−Ban,EG )
rel rel
has small Hom-sets and right derived functors exist.
(ii) On Ban choose the exact structure given by the class E consisting of all
max
kernel-cokernel pairs. It is well-known that (Ban,E ) has enough of both
max
projective and injective objects.
(iii) Choose the left truncation functor τ≤0 on Ch(Ban) given on objects by
(cid:2)
τ≤0E =(···→E−2 →E−1 →Kerd0 →0→···).
(cid:2) E
It induces the left t-structure (D≤0,D≥0) on D+(Ban,E ) whose heart is
(cid:2) (cid:2) max
equivalent to Waelbroeck’s abelian category qBan. Let
H0 :=τ≥0τ≤0 ∼=τ≤0τ≥0
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)
be the associated homological functor H0 : D+(Ban,E ) → qBan and
(cid:2) max
write as usual Hn(E)=H0(E[n]) for n∈Z.
(cid:2) (cid:2)
1. RECONSTRUCTING THE BURGER-MONOD THEORY ix
Remark. For the moment it suffices to know about qBan that it is abelian and
that its objects are represented by pairs (E−1 (cid:4)→ E0) of Banach spaces and an
injective bounded linear map between them. Moreover, there is an embedding (a
fully faithful functor) ι : Ban → qBan given on objects by ι(E) = (0 (cid:4)→ E)
and that ι admits a left adjoint because Ban has cokernels, see Lemma 2.2.6 of
Chapter III.
Letusgiveasimpleproofoftheexistenceofenoughinjectivesin(G−Ban,EG ).
rel
Underlyingtheargument istheconceptofamonad(triple)arisingfromanadjoint
pair of functors, see Weibel [Wei94, Section 8.6].
Recall that a function f : G → E from a Hausdorff topological group G into
a Banach space E is called left uniformly continuous if for every net g → 1 in G
λ
one has sup (cid:5)f(g x)−f(x)(cid:5) −λ−→−−∞→0.
x∈G λ E
Lemma. The forgetful functor ↓: G−Ban → Ban has a right adjoint given on
objects E ∈Ban by the Banach space
Clu(G,E)={f :G→E : f is bounded and left uniformly continuous}
b
equipped with the supremum norm and the G-action (gf)(x)=f(g−1x).
Proof. Let L be the forgetful functor and let R = Clu(G,−). Let M ∈ G−Ban
b
and let E ∈ Ban be any two objects. The unit and counit of the adjunction are
the natural transformations
η :idG−Ban ⇒RL and ε:LR⇒idBan
given by
η (m)=(x(cid:8)→x−1m)∈RL(M)=Clu(G,↓M) and ε (f)=f(1)∈E
M b E
for m∈M and for f∈LR(E)=↓Clu(G,E). Since (cid:5)m(cid:5) =sup (cid:5)η (m)(x)(cid:5) ,
b M x∈G M M
weseethatη (m)isaboundedfunction. Letuscheckthatη (m)isleftuniformly
M M
continuous, so let g →1 be a net and compute for m∈M
λ
sup(cid:5)η (m)(g x)−η (m)(x)(cid:5) = sup(cid:5)x−1g−1m−x−1m(cid:5) =(cid:5)g−1m−m(cid:5)
M λ M M λ M λ M
x∈G x∈G
thelasttermconvergestozerobecausetheactionofGonM isstronglycontinuous.
Moreover, η :M →RL(M) is G-equivariant because
M
η (gm)(x)=x−1gm=(g−1x)−1m=η (m)(g−1x)=[gη (m)](x).
M M M
Thefactthatη andεarenaturaltransformationsisobviousaswellastheonethat
the compositions
L(M)−L−(−ηM−→) LRL(M)−ε−L−(M−→) L(M) and R(E)−η−R−(−E→) RLR(E)−R−(−ε−E→) R(E)
are equal to id and to id . Thus we have L(cid:9)R by [ML98, Ch. IV.1]. (cid:2)
L(M) R(E)
Corollary. There are enough injectives in (G−Ban,EG ).
rel
Proof. Choosethesplit exactstructureE onBansothateveryBanachspace
min
is (projective and) injective. Observe that the forgetful functor
↓:(G−Ban,EG )−→(Ban,E )
rel min