Two-dimensional foliations on four-manifolds [Elektronische Ressource] / Jonathan Bowden

Jonathan Bowden

Sujets

• Mathematics

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2010
• Nombre de lectures: 27
• Langue: English

Extrait

Jonathan Bowden
Two-dimensional foliations on
four-manifolds
Dissertation an der Fakultat fur
Mathematik, Informatik und Statistik der
Ludwig-Maximilians-Universitat Munc hen
vorgelegt am 31. August 2010Erster Gutachter: Prof. Dieter Kotschick, D. Phil. (Oxon)
(Ludwig-Maximilians-Universit at Munc hen)
Zweiter Gutachter: Prof. Dr. Elmar Vogt
(Freie Universit at Berlin)
Termin der mundlic hen Prufung: 17. Dezember 2010Abstract
We study two-dimensional foliations on four-manifolds and examine properties of their closed leaves.
After considering the general case of smooth foliations, we focus on foliations with symplectic leaves
and then on symplectic pairs. In both cases certain restrictions on the underlying distributions and
on the closed leaves of such foliations are derived.
We further study the geometry of characteristic classes of surface bundles with and without at
structures. For general surface bundles we show that the MMM-class are hyperbolic in the sense
of Gromov and deduce certain restrictions on the topology of bundles under the assumption that
the base is a product or that the bundle is holomorphic. We further consider characteristic classes
of at bundles, whose horizontal foliations have closed leaves and compute the abelianisation of
the di eomorphism group of a compact surface with marked points. When the foliations have a
transverse symplectic structure, we show the non-triviality of certain derived characteristic classes
in leaf-wise cohomology. For bundles with boundary we show that there is a relationship between
the geometry of a at structure and the topology of the boundary.
We also introduce the relation of symplectic cobordism amongst transverse knots. Specialising
to the case of symplectic concordance we produce an in nite family of knots that show that this
relation is not symmetric, in stark contrast to its smooth counterpart.
Zusammenfassung
Wir besch aftigen uns mit zwei-dimensionalen Bl atterungen auf Viermanigfaltigkeiten und unter-
suchen die Eigenschaften ihrer abgeschlossenen Bl atter. Nachdem wir den Fall von glatten Bl atter-
ungen betrachtet haben, konzentrieren wir uns auf Bl atterungen mit symplektischen Bl attern und
anschlie end auf symplektische Paare. Fur diese beiden F alle zeigen wir, dass die zugrundeliegen-
den Distributionen und abgeschlossenen Bl atter solcher Bl atterungen gewissen Beschr ankungen
unterliegen.
Weiterhin untersuchen wir die Geometrie der charakteristischen Klassen von Fl achenbundeln
mit und ohne ache Strukturen. Fur allgemeine Flachenbundel zeigen wir, dass die MMM-Klassen
hyperbolisch im Sinne von Gromov sind. Unter der Annahme, dass die Basis ein Produkt oder
das Bundel holomorph ist, leiten wir au erdem gewisse Einschr ankungen an die Topologie solcher
Bundel her. Des weiteren behandeln wir charakteristische Klassen acher Bundel, deren horizon-
tale Bl atterungen abgeschlossene Bl atter besitzen und berechnen die Abelianisierung der Di eo-
morphismengruppe einer kompakten Fl ache mit markierten Punkten. Wenn die Bl atterungen eine
transversale symplektische Struktur aufweisen, zeigen wir, dass gewisse sekund are charakteristische
Klassen in der blattweisen Kohomologie nicht trivial sind. Fur Bund el mit Rand leiten wir eine
Beziehung zwischen der Geometrie einer achen Struktur und der Topologie des Randes her.
Schlie lich fuhren wir die Relation des symplektischen Kobordismus fur transversale Knoten
ein. Im Spezialfall der symplektischen Konkordanz zeigen wir mittels einer unendlichen Familie
von Knoten, dass diese Relation im Gegensatz zur glatten Konkordanz nicht symmetrisch ist.Contents
1 Introduction 7
1.1 Foliations and distributions on 4-manifolds . . . . . . . . . . . . . . . . . . . 7
1.2 Surface bundles and their characteristic classes . . . . . . . . . . . . . . . . . 9
1.3 Flat surface bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Surface bundles and extended Hamiltonian groups . . . . . . . . . . . . . . . 12
1.5 Characteristic classes of symplectic foliations . . . . . . . . . . . . . . . . . . 13
1.6 Symplectic cobordism and transverse knots . . . . . . . . . . . . . . . . . . . 13
2 Distributions and leaves of foliations 15
2.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The distribution equations on a 4-manifold . . . . . . . . . . . . . . . . . . . 15
2.3 The Milnor-Wood inequality and compact leaves . . . . . . . . . . . . . . . . 22
2.4 Special classes of foliations and their closed leaves . . . . . . . . . . . . . . . 25
2.5 Symplectic pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1 Topological constructions of symplectic pairs . . . . . . . . . . . . . . 30
2.5.2 Geometry of leaves of symplectic pairs . . . . . . . . . . . . . . . . . 33
3 Surface Bundles and their characteristic classes 37
3.1 Surface bundles and holonomy representations . . . . . . . . . . . . . . . . . 37
3.1.1 Boundedness of the vertical Euler class . . . . . . . . . . . . . . . . . 40
3.1.2 Bounds on self-intersection numbers of multisections . . . . . . . . . 44
3.2 MMM-classes vanish on amenable groups . . . . . . . . . . . . . . . . . . . . 49
3.3 are hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Holomorphic surface bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Flat surface bundles 63
4.1 Closed leaves and horizontal foliations . . . . . . . . . . . . . . . . . . . . . 63
+4.1.1 Computation of H (Diff ( )) . . . . . . . . . . . . . . . . . . . . 681 h;k
c 24.1.2 of H (Diff (R ; 0)) . . . . . . . . . . . . . . . . . . . . 701
4.2 Closed leaves of at bundles with symplectic holonomy . . . . . . . . . . . . 71
4.2.1 The case of genus 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Surface bundles and extended Hamiltonian groups 79
15.1 Filling at S -bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Flat bundles and the extended group . . . . . . . . . . . . . . . 82
56 Contents
5.3 The Calabi map and the rst MMM-class . . . . . . . . . . . . . . . . . . . . 89
g5.4 The second MMM-class vanishes on Ham . . . . . . . . . . . . . . . . . . . 92
6 Characteristic classes of symplectic foliations 95
6.1 Factorisation of Pontryagin classes . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Gelfand-Fuks cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Characteristic classes of leaves . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7 Symplectic cobordism and transverse knots 109
7.1 Contact manifolds and symplectic cobordisms . . . . . . . . . . . . . . . . . 109
7.2 Symplectic spanning surfaces and slice genus . . . . . . . . . . . . . . . . . . 113
37.3 links are quasipositive in S . . . . . . . . . . . . . . . . . . . . . 115
7.4 Branched covers and symplectic spanning surfaces . . . . . . . . . . . . . . . 117
7.5 Symplectic concordance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A Five-term exact sequences 123
Bibliography 127Chapter 1
Introduction
The motivating theme of this thesis is the study of 2-dimensional foliations on 4-manifolds.
In studying examples of such foliations the geometry of surface bundles consequently took
on a prominent role and hence the majority of results we present are in some way related to
surface bundles. Another recurring theme is that of 4-dimensional symplectic topology and
as such we also consider the topology of certain symplectic surfaces in the form of cobordisms
of transverse knots in contact manifolds.
1.1 Foliations and distributions on 4-manifolds
The question of whether a given manifold admits a foliation of a given dimension is a very
di cult one in general. An obvious necessary condition for the existence of a q-dimensional
foliation is the existence of a q-dimensional distribution. In dimensions 1 and 2 this is
in fact also su cient. For dimension 1 this is obvious and in dimension 2 it follows from
Thurston’s h-principle, which says that on a manifold of dimension at least 4, any oriented
2-dimensional distribution is homotopic to an integrable one (cf. [Th2]). Thus the existence
of foliations reduces to the problem of the existence of oriented 2-plane distri-
butions. An oriented 2-plane eld is then a section of the Grassmanian bundle of 2-planes
and the existence of such sections is a homotopy problem that can be expressed in terms of
obstruction theory.
In the case of oriented 4-manifolds the existence of an oriented 2-dimensional distribution
is equivalent to a splitting of the tangent bundle as the Whitney sum of two oriented rank-2
subbundles ; . As these bundles are oriented there is an almost complex structure on1 2
TM so that both bundles are complex subbundles. Thus the Whitney sum formula yields
certain equations that the Chern classes of ; must satisfy. Since the Euler class of an1 2
oriented rank-2 bundle is the same as its rst Chern class, these equations may be written
as follows:
2 2 2e( )‘e( ) =c (M) and e ( ) +e ( ) =c (M):1 2 2 1 2 1
Another necessary condition that these classes satisfy is that e( ) +e( ) reduce to the1 2
second Stiefel-Whitney class w (M) in mod 2 cohomology. The latter condition combined2
with the above equations give what we shall call the distribution equations. It is an old result
78 1. Introduction
of Hirze