Topological properties of asymptotic invariants and universal volume bounds [Elektronische Ressource] / Michael Brunnbauer

Michael Brunnbauer

Sujets

• Mathematics

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2008
• Nombre de lectures: 11
• Langue: English

Extrait

Michael Brunnbauer
Topological properties of
asymptotic invariants and
universal volume bounds
Dissertation an der Fakult¨at fur¨
Mathematik, Informatik und Statistik der
Ludwig-Maximilians-Universit¨at Munc¨ hen
vorgelegt am 9. Mai 2008Erster Gutachter: Prof. Dieter Kotschick, D. Phil. (Oxon)
(Ludwig-Maximilians-Universit¨at Munc¨ hen)
Zweiter Gutachter: Prof. Mikhael Gromov, Ph. D.
´(Institut des Hautes Etudes Scientifiques;
Courant Institute of Mathematical Sciences)
Dritter Gutachter: Prof. Dr. Thomas Schick
(Georg-August-Universit¨at G¨ottingen)
Termin der mundlic¨ hen Pruf¨ ung: 11. Juli 2008Abstract
In this thesis, we prove that many asymptotic invariants of closed manifolds de-
pend only on the image of the fundamental class under the classifying map of the
universal covering. Examples include numerical invariants that reflect the asymp-
totic behaviour of the universal covering, like the minimal volume entropy and
the spherical volume, as well as properties that are qualitative measures for the
largeness of a manifold and its coverings, like enlargeability and hypersphericity.
Another important class of invariants that share the above invariance prop-
erty originates from universal volume bounds. The main example is the systolic
constant, which encodes the relation between short noncontractible loops and the
volume of a manifold. Further interesting examples are provided by the optimal
constantsinGromov’sfillinginequalities,forwhichweshowthattheydependonly
on the dimension and orientability.
Consideringhigher-dimensionalgeneralizationsofthesystolicconstant,acom-
plete answer to the question about the existence of stable systolic inequalities is
given. In the spirit of the results mentioned already, we also prove that the stable
systolic constant depends only on the image of the fundamental class in a suitable
Eilenberg-MacLane space.
Zusammenfassung
In dieser Arbeit wird gezeigt, dass viele asymptotische Invarianten geschlossener
Mannigfaltigkeiten nur vom Bild der Fundamentalklasse unter der klassifizieren-
¨den Abbildung der universellen Uberlagerung abhangen.¨ Hierzu z¨ahlen sowohl
¨numerische Invarianten, die das asymptotische Verhalten der universellen Uberla-
gerung widerspiegeln, wie die minimale Volumenentropie und das sph¨arische Vo-
lumen, als auch Eigenschaften, die qualitative Maße fur¨ die Gr¨oße einer Mannig-
¨faltigkeit und ihrer Uberlagerungen darstellen, wie Vergr¨oßerbarkeit und Hyper-
sph¨arizit¨at.
EineweiterewichtigeKlassevonInvarianten,diedieobigeInvarianzeigenschaft
teilen, erh¨alt man aus universellen Volumenschranken. Das wichtigste Beispiel
hierfur¨ ist die systolische Konstante, die das Verh¨altnis zwischen kurzen nichtzu-
sammenziehbaren Schleifen und dem Volumen einer Mannigfaltigkeit wiedergibt.
Weitere interessante Beispiele werden durch die optimalen Konstanten in Gro-
movs Filling-Ungleichungen gegeben, von denen gezeigt wird, dass sie nur von der
Dimension und der Orientierbarkeit abhangen.¨
BeiderBetrachtunghoher-dimensional¨ erVerallgemeinerungendersystolischen
Konstante wird eine vollst¨andige Antwort auf die Frage nach der Existenz stabiler
systolischer Ungleichungen gefunden. In Analogie zu den oben erw¨ahnten Ergeb-
nissen wird bewiesen, dass die stabile systolische Konstante nur vom Bild der
Fundamentalklasse in einem passenden Eilenberg-MacLane-Raum abhangt.¨Contents
1 Introduction 7
1.1 Asymptotic invariants . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Systolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Further universal volume bounds . . . . . . . . . . . . . . . . 13
1.4 Largeness properties . . . . . . . . . . . . . . . . . . . . . . . 14
2 Homologicalinvarianceforasymptoticinvariantsandsystolic
inequalities 19
2.1 Topological preliminaries . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 The Hopf trick . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Orientation issues . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Maps to n-dimensional CW complexes . . . . . . . . . 24
2.1.4 Maps to arbitrary CW complexes . . . . . . . . . . . . 26
2.1.5 Absolute and geometric degree. . . . . . . . . . . . . . 28
2.2 Systolic constants and minimal volume entropy . . . . . . . . 30
2.2.1 Systolic constants . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Minimal volume entropy . . . . . . . . . . . . . . . . . 32
2.2.3 Comparison axiom and homotopy invariance . . . . . . 33
2.2.4 Extension axiom . . . . . . . . . . . . . . . . . . . . . 36
2.2.5 Systolic manifolds . . . . . . . . . . . . . . . . . . . . . 37
2.3 Spherical volume . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Homological invariance and first applications . . . . . . . . . . 42
2.4.1 Homological invariance . . . . . . . . . . . . . . . . . . 43
2.4.2 Degree one maps . . . . . . . . . . . . . . . . . . . . . 45
2.4.3 Adding simply-connected summands . . . . . . . . . . 46
2.4.4 Z -systoles . . . . . . . . . . . . . . . . . . . . . . . . . 472
2.5 An inequality between the systolic constant and the minimal
volume entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5.1 Strong extension axiom . . . . . . . . . . . . . . . . . . 48
2.5.2 Geometric cycles . . . . . . . . . . . . . . . . . . . . . 54
56 Contents
3 Filling inequalities do not depend on topology 59
3.1 Optimal filling ratios . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.1 Filling radius and filling volume . . . . . . . . . . . . . 62
3.1.2 Axioms for filling ratios . . . . . . . . . . . . . . . . . 66
3.2 Constancy of the optimal filling ratios. . . . . . . . . . . . . . 72
4 On manifolds satisfying stable systolic inequalities 77
4.1 On higher-dimensional systoles . . . . . . . . . . . . . . . . . 81
4.1.1 Higher-dimensional systolic constants . . . . . . . . . . 81
4.1.2 Axioms for stable constants . . . . . . . . . . . 83
4.2 Stable systolic constants . . . . . . . . . . . . . . . . . . . . . 87
4.2.1 Spheres and their loop spaces . . . . . . . . . . . . . . 87
4.2.2 Existence of stable systolic inequalities . . . . . . . . . 89
4.2.3 Homological invariance for stable systolic constants . . 92
4.2.4 The multilinear intersection form . . . . . . . . . . . . 94
5 Enlargeability is homologically invariant 97
5.1 Large Riemannian manifolds . . . . . . . . . . . . . . . . . . . 99
5.2 Essentialness and homological invariance . . . . . . . . . . . . 107
5.2.1 Large homology classes . . . . . . . . . . . . . . . . . . 107
5.2.2 Higher enlargeability implies essentialness . . . . . . . 112
Bibliography 115
Index 123Chapter 1
Introduction
Thisthesisisconcernedwithvariousinvariantsandpropertiesofclosedman-
ifolds that describe asymptotic aspects of the universal covering or that arise
as optimal constants in curvature-free bounds on the volume. We will prove
that many of these invariants depend only on the image of the fundamental
classundertheclassifyingmapoftheuniversalcovering. Thisbehaviourwill
be called homological invariance.
Examples for asymptotic invariants include the minimal volume entropy,
whichdescribestheexponentialvolumegrowthoftheuniversalcovering,and
the spherical volume, which is defined via immersions of the universal cover-
ingintothespaceofsquare-integrablefuntions. Moreover,enlargeabilityand
manyotherlargenesspropertiesoftheuniversalcoveringlikehypersphericity
or macroscopic largeness are also homologically invariant in the above sense.
Universal volume bounds like Gromov’s celebrated systolic and filling
inequalities define, via the optimal constants in these inequalities, numerical
invariantsofmanifolds. Themostprominentexampleisthesystolicconstant,
which determines the relation between the length of short noncontractible
loops and the volume of the manifold. As an application of our results on
homological invariance, an inequality between the minimal volume entropy
and the systolic constant is derived.
The invariants mentioned so far actually depend on the fundamental
group, that is, their values change if the fundamental group changes. The
optimal constants in Gromov’s filling inequalities do not. More precisely, we
will show that they depend only on the dimension and orientability.
A slightly different kind of curvature-free volume bound is provided by
the stable systolic inequalities. Here, the lower bound on the volume is not
given by a one-dimensional quantity like the length of noncontractible loops
butbythestabilizedvolumeofhigher-dimensionalsubmanifoldsthatarenot
nullhomologous. Therefore, it is not surprising that homological invariance
78 1. Introduction
holds only after replacing the classifying map of the universal covering by a
suitablemaptoan Eilenberg-MacLane space of higher degree. Moreover, we
willgiveacompleteanswertothefreedomproblemofstablesystoles, thatis,
we will find a topological characterization for the existence and nonexistence
of stable systolic inequalities.
In the following sections, we will give more details on the definitions of
the invariants mentioned above, try to motivate our interest in them, and
present the main results of the thesis. Some of these results are not stated
in full