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L_1hn2-Betti numbers of R-spaces and the integral foliated simplicial volume [Elektronische Ressource] / vorgelegt von Marco Schmidt

107 pages
Marco Schmidt2L -Betti Numbers of R-Spaces and theIntegral Foliated Simplicial Volume2005Mathematik2L -Betti Numbers ofR-Spaces and theIntegral Foliated Simplicial VolumeInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch-Naturwissenschaftlichen Fakult¨atder Westf¨alischen Wilhelms-Universit¨at Munster¨vorgelegt vonMarco Schmidtaus Berlin– 2005 –Dekan: Prof. Dr. Klaus HinrichsErster Gutachter: Prof. Dr. Wolfgang L¨ uckZweiter Gutachter: Prof. Dr. Anand DessaiTag der mundlichen¨ Pr¨ufung: 30. Mai 2005Tag der Promotion: 13. JuliIntroductionThe origin of this thesis is the following conjecture of Gromov [26, Section 8A,(2)2p. 232] revealing a connection between the L -Betti numbers b (M)and the sim-kplicial volumeM ofaclosedorientedconnectedasphericalmanifold M.Conjecture. Let M be a closed oriented connected aspherical manifold with M = 0.Then(2)b (M)=0 forall k≥ 0.k2The first definition of L -Betti numbers for cocompact free proper G-manifoldswith G-invariant Riemannian metric (due to Atiyah [2]) is given in terms of theheat kernel. Wewill briefly recall this original definitionatthebeginningofChap-2ter 1. Today, there is an algebraic and more general definition of L -Betti numberswhich works for arbitrary G-spaces. Analogously to ordinary Bettinumbers, they2are given as the “rank” of certain homology modules.
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Marco Schmidt
2L -Betti Numbers of R-Spaces and the
Integral Foliated Simplicial Volume
2005Mathematik
2L -Betti Numbers ofR-Spaces and the
Integral Foliated Simplicial Volume
Inaugural-Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften im Fachbereich
Mathematik und Informatik
der Mathematisch-Naturwissenschaftlichen Fakult¨at
der Westf¨alischen Wilhelms-Universit¨at Munster¨
vorgelegt von
Marco Schmidt
aus Berlin
– 2005 –Dekan: Prof. Dr. Klaus Hinrichs
Erster Gutachter: Prof. Dr. Wolfgang L¨ uck
Zweiter Gutachter: Prof. Dr. Anand Dessai
Tag der mundlichen¨ Pr¨ufung: 30. Mai 2005
Tag der Promotion: 13. JuliIntroduction
The origin of this thesis is the following conjecture of Gromov [26, Section 8A,
(2)2
p. 232] revealing a connection between the L -Betti numbers b (M)and the sim-k
plicial volumeM ofaclosedorientedconnectedasphericalmanifold M.
Conjecture. Let M be a closed oriented connected aspherical manifold with M = 0.
Then
(2)
b (M)=0 forall k≥ 0.k
2The first definition of L -Betti numbers for cocompact free proper G-manifolds
with G-invariant Riemannian metric (due to Atiyah [2]) is given in terms of the
heat kernel. Wewill briefly recall this original definitionatthebeginningofChap-
2ter 1. Today, there is an algebraic and more general definition of L -Betti numbers
which works for arbitrary G-spaces. Analogously to ordinary Bettinumbers, they
2are given as the “rank” of certain homology modules. More precisely, the k-th L -
(2)Bettinumberb (Z;NG)ofaG-space ZisthevonNeumanndimensionofthek-thk
twistedsingularhomologygroupofZwithcoefficientsinthegroupvonNeumann
algebra NG. Here, von Neumann dimension means the dimension function de-
veloped by Lu¨ck [36],[37] for arbitrary modules (in the algebraic sense) overfinite
(2)2von Neumann algebras. The k-th L -Betti number b (G) of a group G is definedk
(2)as b (EG;NG),whereEG → BG is the universal principal G-bundle. The nota-k

(2) (2)
tion b (M) isshortfor b M;N π (M) . Wewill presenttherelevant definitions1k k
2in Chapter 1. The standard reference for L -Bettinumbersis Luck’s¨ extensivetext-
book[38].
The simplicial volume M is a real valued homotopy invariant for closed ori-
entedconnectedtopologicalmanifolds M. Itmeasuresthe“complexity”ofthefun-
damental homology class of M.Namely,M is defined as the infimum of the
1 -norms of real singular cycles representingthefundamental class. Thesimplicial
volume wasdefinedbyGromovin ordertogive volumeestimatesforRiemannian
manifolds [24]. Inparticular, Gromovwasinterestedinlowerboundsforthemini-
malvolumeminvol(M),ifMisinadditionsmooth. Definitionandpropertiesofthe
simplicial volume andtherelation totheminimal volume are treatedin Chapter2.
It is quite interesting that there are many connections to Riemannian geometry al-
though the definition of simplicial volume only takes the topological structure of
the manifold into account. For example, a manifold with vanishing simplicial vol-
umecannotcarryaRiemannianmetricofnegativesectionalcurvature[29].
v2At first sight the definitions of L -Betti numbers and the simplicial volume do
not indicate a relationship between them, but in certain situations both invariants
behave similarly. We will provide examples for this in Section 5.1. These similari-
2tiessuggestaconnectionbetween L -Bettinumbersand thesimplicial volume. An
immediate consequence of Gromov’s Conjecture would be the fact that the Euler
characteristic χ(M) vanishesif M is asphericalwithM = 0. Thisfact couldhave
2been the motivation for Gromov to study relations between L -Betti numbers and
2simplicial volume. Gromov’s Conjecture would also imply that the L -Betti num-
bersoftheuniversalcovering M ofan asphericalmanifold M vanish if M admitsa
selfmap M → M ofdegree d∈{/ −1,0,1}.ThiswasprovedbyLuc¨kunderthe ad-
ditional assumption that each normal subgroup offinite index of the fundamental
group π (M) is Hopfian[38, Theorem14.40, p. 499]. A group G is called Hopfian if1
eachsurjectivegrouphomomorphismG→ G isanisomorphism.
2In this thesis, we will pick another approach to L -Betti numbers of G-spaces.
Wefollow thephilosophythat it is oftenfruitful tolook at thesameinvariant from
different points of view. A basic example for this is provided by the Betti num-
bers b (Z) offinite CW-complexes Z: Looking only at the singular chain complex,k
it is not clear that they are finite. Looking only at the cellular chain complex, one
does not see directly that they are independentof the CW-structure. But once one
has shown that both chain complexes have isomorphic homology groups, it turns
out that the b (Z) are homotopy invariants with values in the nonnegative inte-k
2gers.WewilseethatL -Betti numbers are another good instance for this general
principle. For some properties the original analytic approach of Atiyah is more
2convenient(e.g.,thevanishingof L -Bettinumbersoftheuniversalcoveringofhy-
perbolic manifolds outside the middle dimension [16]), other properties are more
2easily proved with Lu¨ck’s algebraic definition, such as the fact that L -Betti num-
bersofuniversalcoveringsofasphericalspaceswithamenablefundamentalgroup
vanish [37, Section 5, p. 155 ff.]. In this thesis we will give another definition of
2L -Bettinumberswhichisadequateforanalyzing Gromov’sConjecture.
2We defineL -BettinumbersofR-spaces. Thesearespaceswhichareparametrized
overastandardBorelprobabilityspaceXandprovidedwithanactionofastandard
equivalence relationR⊂X×X. The starting point of exploring standard equiva-
lencerelationsistheworkofFeldmanandMoore[18]. ThedefinitionofR-spacesis
duetoConnes. Thebasicexampleforstandardequivalencerelationsisgivenbythe
orbitequivalencerelationR ofastandardaction G X. Eachstandardequiv-GX
alencerelationisanorbitequivalencerelationofanappropriatemeasurepreserving
action G X [18], but one cannot assume that the action is essentially free. The
2definition of L -Betti numbers of R-spaces makes use of the equivalence relation
ringZR and the equivalence relation von Neumann algebraNR.Withthesetwo
objectsreplacingthegroupringandthegroupvonNeumannalgebra,thedefinition
2isanalogoustothealgebraic definitionof L -BettinumbersofG-spaces. Infact, we
Xfirst define a singular chain complex C (S;Z) for anR-space S and consider then

X 2the homology of the complexNR⊗ C (S;Z).Thek-th L -Betti number of S is
ZR

videfinedas
Xdim H NR⊗ C (S;Z) .
NR ZRk •
If G X isastandardaction,thereisaninductionfunctor
ind: G-Spaces→R -SpacesGX
which sends a G-space Z to the R -space X× Z.InChapter4,weshowtheGX
followingresult.
Theorem. Let G be a countable group and G X a standard action on a standard Borel
space X. Thenfor acountable free G-CW-complex Z onehas
(2) (2)b (X×Z;NR )=b (Z;NG) for all k≥ 0.GXk k
ThisisaslightgeneralizationofaresultofGaboriau whoprovedthesameresult
(2)2for countable free simplicial complexes. He also defined L -Bettinumbers b (R)k
(2) (2)of a standard equivalence relationRandprovedthatb (R )=b (G) holdsGXk k
2forastandardaction G X. Gaboriau usedthis resultto showthat L -Bettinum-
bers of orbit equivalent groups coincide and those of measure equivalent groups
coincide up to a non-zero multiplicative constant. Orbit equivalence will be in-
troduced in Section 4.6, where we also give a new proof for the orbit equivalence
2invariance of L -Bettinumbers. The definition of measure equivalence, which can
be viewed as a measure theoretic analogue of quasi isometry, is due to Gromov
andZimmer. AlotofworkonmeasureequivalencewasdonebyFurman[19],[20].
It should be mentioned that Sauer [41] reproved Gaboriau’s theorem using the di-
mension theory of Lu¨ck. More information about about measure equivalence can
befoundin[22],[41].
2The first motivation for the approach to L -Betti numbers via the detour to R-
spaceswasaremarkofGromov[27,p.306f.],indicatinghowonecouldtrytoprove
2the conjecture relating L -Betti numbers and the simplicial volume of aspherical
manifolds. Thestartingpointforthatistheupperbound
n
(2) n+1
b (M)≤ 2 · M ,
Z
∑ j
j=0
whereM denotestheintegralsimplicialvolumewhichisgivenastheminimum
Z
1of -normsofintegral fundamentalcycles. Thisinequalityisaneasyapplication of
the Poincare´ duality theorem. Unfortunately,M is only a very rough estimate
Z
for the simplicial volume M (e.g., M ≥ 1 holds for all manifolds M), and
Z
2thePoincare´ dualityargumentdoesnotprovideanupperboundforthesumof L -
Bettinumbersintermsofthecoefficientsofa real fundamentalcycle(andtherefore
of the simplicial volume). Gromov’s idea behind the integral foliated simplicial vol-
umeM is to resolve this drawback by introducing weighted singular cycles to
F,Z
representthefundamentalclassof M. Theweightisgivenbythecoefficientswhich
vii∞are functions f ∈ L (X;Z). One has the inequality M ≤ M .Roughly
F,Z Z
speaking,theintegralfoliatedsimplicialvolumeisgivenastheinfimumofthesum

1of -norms |f| of the coefficient functions f of “weighted” fundamental cycles.X
This definition is only vaguely indicated by Gromov [27, p. 305f.]. The following
n n+1theoremisposedthereasanexercise(with2 insteadof2 asconstantfactor):
Theorem. Let M beaclosed connectedoriented manifold ofdimension n. Then
n
(2) n+1
b (M)≤ 2 · M
∑ F,Zj
j=0
holds.
InChapter5,wegiveaconcisedefinitionofM andprovethistheorem.
F,Z
Note that there is no asphericity condition in the theorem. In order to prove
Gromov’sconjecture,onecouldtrytoprovethatM = 0impliesM =0for
F,Z
aspherical manifolds M. Unfortunately, we are far away from such a result. The
1bestoutcomeinthisdirectionisS × M = 0.
F,Z
Actually, the current definition of M and the proof of the above theorem
F,Z
does not make use of R-spaces. Hence the thesis somehow breaks up into two
2independent parts, one about R-spaces and their L -Betti numbers and one deal-
ing with Gromov’s conjecture. There is a version ofM ,whereR-spaces and
F,Z
2their L -Bettinumbersoccur,butthisversionneedsthefactthatthecorresponding
cohomology

•H hom C (S;Z),NR
ZR •
satisfies Eilenberg-Steenrod-typeaxioms, and we could not prove the excision ax-
iom. Nevertheless,thedefinitionofM ismotivatedbythestudyofR-spaces
F,Z
andtheirhomology.
This thesis is organized as follows: The first two chapters consist of a survey
2about the concepts appearing in Gromov’s Conjecture, namely L -Betti numbers
andsimplicial volume.
2Chapter 1 deals with L -Betti numbers of G-spaces. We describe the algebraic
2approach to L -Bettinumbers using Lu¨ck’s dimension theory and collect the main
propertieswithoutproof.
In Chapter 2, the simplicial volume of closed oriented connected manifolds is
introduced. In addition to the definition and properties, we describe bounded co-
homologyasamain toolforexploringsimplicial volume. Asectionaboutminimal
volume is included since the study of this invariant was the main motivation for
Gromovtoanalyzesimplicial volume.
2Chapters3and4containtheapproachto L -BettinumbersviaR-spaces.
In Chapter 3, we presentthe definitionofR-spaces. We examine standard Borel
spacesandassigntoastandardBorelspaceXwithprobabilitymeasurethecategory
viii

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