FOLIATIONS 2005
editors
Pawel Walczak Remi Langevin Steven Hurder Takashi Tsuboi
FOLIATIONS 2005
FOLIATIONS 2005 Proceedings of the International Conference Lodz, Poland 13 - 24 June 2005
editors P a w e l Walczak Uniwersytet Lodzki, Lodz, Poland
Remi Langevin Universite de Bourgogne, Dijon, France
Steven H u r d e r University of Illinois at Chicago, USA
Takashi Tsuboi University of Tokyo, Japan
\jjp World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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PREFACE This volume contains a set of papers written by the participants of the international conference Foliations 2005 held in the Conference Center of the University of Lodz (Poland) between June 13 and June 24, 2005. The conference was sponsored by the Chair of Geometry and Faculty of Mathematics of the University of Lodz, the Japan Society for Promotion in Science, Ministry of Education and Sport of the Republic of Poland and the Stefan Banach International Mathematical Center in Warsaw. Participants numbered in approximately 75, more than 50 of them from abroad (Austria, Brazil, France, Germany, Japan, Korea, Romania, Russia, Spain, Tunisia, Ukraine and USA). The organizing committee consisted of four people who became editors of this volume. They were supported by the staff of the Faculty of Mathematics and a group of young mathematicians and students from the University of Lodz. The papers contained in this volume are closely related to the lectures given at the conference, which was designed to cover various aspects of the theory of foliations, focusing on topology, geometry and dynamics of such objects. All the papers contained in this volume were refereed by experts. Some of them are just research papers containing original results, some contain surveys which bring light to the current state of some aspects of the foliation theory, some are of "mixed-type" (surveys with a bit of new results). The volume contains also a list of open problems presented during the conference at the problem session, then collected and prepared for publication by one of the editors of the volume. We hope that both, the conference itself and this volume of proceedings, should make a significant contribution to the progress of our field of science. We express our gratitude to the participants, the contributors of this volume, the sponsors and all the colleagues and students who helped us while organizing the conference and preparing the volume for publication. In particular, we would like to mention Marek Badura (who organized the webpage of the conference as well as the participant data base), Maciej Czarnecki (the secretary of the organizing committee) and Zona Walczak (who worked a lot with the TeX-files of the articles contained here). The Editors
CONTENTS J. Alvarez Lopez, X. Masa: Morphisms of pseudogroups and foliated maps
1
M. Asaoka: Codimension-one foliations with a transversely contracting flow
21
T. Asuke: On infinitesimal derivatives of the Bott class A. Ayadi, H. Marzougui: Dense orbits for abelian subgroups of GL{n, C) A. Bis, S. Hurder, J. Shive: Hirsch foliations in codimension greater than one
37
47 71
D. Bolotov: Extrinsic geometry of foliations on 3-manifolds
109
C. Bonatti, V. Grines, O. Pochinka: Classification of MorseSmale diffeomorphisms with the chain of saddles on 3-manifolds
121
M. Czarnecki, P. Walczak: Extrinsic geometry of foliations
149
V. Grines, E. Zhuzhoma: Surface dynamical systems and foliations via geodesic laminations
169
T. Inaba: On rigidity of submanifolds tangent to nonintegrable distributions
203
S. D. Jung: Transversal twistor spinors on a Riemannian foliation
215
S. Kamatani, H. Kodama, T. Noda: A Birkhhoff section for the Bonatti-Langevin example of Anosov flow
229
F. Kopei: A remark on a relation between foliations and number theory
245
vii
Vlll
Y. Kordyukov: Noncommutative spectral geometry of Riemannian foliations: Some results and open problems
251
T. Kuessner: A survey on simplicial volume and invariants of foliations and laminations
293
R. Langevin: Harmonic foliations of the plane, a conformal approach
315
S. Maksymenko: Consecutive shifts along orbits of vector fields
327
K. Mikami, T. Mizutani: A Lie algebroid and a Dirac structure associated to an almost Dirac structure
341
Y. Mitsumatsu: Convergence of contact structures to foliations K. Richardson: Generalized equivariant index theory V. Slesar: Vanishing results for spectral terms of a Riemannian foliation D . Toben: The generalized Weyl group of a singular Riemannian foliation T. Tsuboi: On the group of foliation preserving diffeomorphisms
353 373
389 399
411
P. Walczak: Conformally defined geometry on foliated Riemannian manifolds
431
S. Hurder: Problem Set
441
List of participants
477
Program
479
FOLIATIONS 2005 ed. by Pawei W A L C Z A K et al. World Scientific, Singapore, 2006 pp. 1-19
M O R P H I S M S OF P S E U D O G R O U P S A N D FOLIATED M A P S JESUS A. ALVAREZ LOPEZ XOSE M. MASA Departamento de Xeometria e Topoloxia Facultade de Matemdticas Universidade de Santiago de Compostela 15782 Santiago de Compostela, Spain, e-mail:
[email protected], e-mail:
[email protected] 1
Introduction
In [2], we have introduced the concept of morphism of pseudogroups generalizing the etale morphisms of Haefliger [11]. With our definition, any continuous map between foliated spaces mapping leaves to leaves (a foliated map) induces a morphism between the corresponding holonomy pseudogroups. This concept can be interpreted as a morphism of S-atlases, defined by Van Est [17], and is more general than a homomorphism of etale groupoids: only transverse foliated maps induce homomorphisms of holonomy groupoids. The main result of [2] states that any morphism between complete pseudogroups of local isometries is complete, has a closure and its maps are C°° along the orbit closures. Here, completeness and closure are obvious versions for morphisms of concepts introduced by Haefliger for pseudogroups. The proof of this theorem only involves basic techniques, but it is rather complicated. It is applied to approximate foliated maps by smooth ones in the case of complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. The goal of this paper is to clarify that theorem by giving a simplified proof for the case of dense orbits, recalling the main ideas without many 1
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details. Applications, examples and related open problems are also given. 2
Morphisms of pseudogroups
A pseudogroup ft of local transformations of a space T (or acting on T) is a collection of homeomorphisms between open subsets of T which contains the identity map idf, and is closed under the operations of composition (wherever defined), inversion, restriction to open sets and combination [11, Section 1.4]. The pseudogroup ft is generated by a subset S C ft if any map in ft can be obtained from S by using these operations. The restriction of H t o a subspace To C T is the pseudogroup H\T0 of local transformations of To that can be locally extended to maps in ft. If T is a C°° manifold and the maps in ft are C°°, then ft is called C°°. The basic dynamical concepts can be generalized to pseudogroups: orbits, saturations, invariant or saturated sets, etc. The quotient space of ft-orbits is denoted by ft\T. For any open U C T, let Hv = {ft G ft | dom/i = U). Let ft and ft' be pseudogroups acting on spaces T and T". According to [11], an etale morphism $ : ft —> ft' is a maximal collection of homeomorphisms of open subsets of T to open subsets of T" satisfying the following properties. (i) If / G $, ft G ft and ft' G ft', then /i' o / o ft e $. (ii) The domains of elements of $ cover T. (iii) If <j), tp G $, then V ° ^ _ 1 G W. If $ _ 1 = {_1 | 0 G $ } is an etale morphism too, then $ is called an equivalence and W is said to be equivalent to H'. In this case, 7i" = H u H ' U $ U $ _ 1 is a pseudogroup acting on the topological sum T" = TUT' whose orbits cut T and T', and so that H"\T = W and W"| T ' = W. Reciprocally, if a pseudogroup H." acting on T" satisfies these conditions, then {(f> GH" I dom, ip G $o, h £ S and £ G dom fl dom(^ o /i), then there is some h' € TL' with (a:) e dom/i' and so that /i' o <j> = ijj o h on some neighborhood of x. Then there is a unique morphism Q : TL ^> H' containing $o> which is said to be generated by $o- Observe that morphisms consisting of local homeomorphisms are precisely those generated by etale morphisms. The composition of two consecutive morphisms is the morphism generated by the composites of the corresponding maps (wherever defined). With this operation, these morphisms form a category PsGr, whose isomorphisms are the morphisms generated by equivalences. Notice that idy generates the identity morphism id« at TL in PsGr, and H C id^- The restriction of a morphism $ : TL —> TL' to a subspace To C T is the morphism ®\T0 : TL\T0 —* H' consisting of all maps of open subsets of To to T' that can be locally extended to maps in $ . The inclusion map To °—• T generates a morphism TC\T0 —* Tt, whose composition with any morphism H'i, i = 1,2, is the morphism $ ! x $ 2 : Hi x Tt2 -> Wi x W2 generated by the products 0i x 0 2 with 0i € ? ^ , i = 1,2, is the morphism ($1, $2) : W —> H'i x W2 generated by the pairs ((pi, fa), where fa € 3>i and 02 € $2 have the same domain. Let Top be the category of continuous maps between topological spaces. There is a canonical injective covariant functor Top —> PsGr which assigns the pseudogroup generated by idy to each space T, and assigns the morphism generated by / to each continuous map / . We will consider Top as a subcategory of PsGr in this way. Many topological concepts can be generalized to pseudogroups by using morphisms and orbits instead of continuous maps and points: path connectedness, homotopies, homotopy equivalences, singular (co)homology, homotopy groups (see [10, 17]), etc.
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Holonomy pseudogroups of foliated spaces
A foliated structure T of dimension n G N on a space X can be described by a defining cocycle [12], which is a collection {Ui,pi}, where {Ui} is an open cover of X and each Pi is a topological submersion of Ui onto some space Ti whose fibers are connected open subsets of t n , such that the following compatibility condition is satisfied: for every x £ UidUj, there is an open neighborhood C/f ,• of x in Ui C\ Uj and a homeomorphism hf • : PiiUfj) —• Pj(Ufj) so that Pj = hfj opi on Ufj. Another defining cocycle {U'a,p'a} determines the same foliated structure when {Ui,pi} U {U'a,p'a} is a defining cocycle. The space X endowed with T is called a foliated space. The usual terminology of foliations can be generalized to foliated spaces: foliated chart, foliated atlas, plaques, leaf topology, leaves, local transversals, simple open sets, etc. The quotient space of leaves is denoted by XjT. Notice that T can be identified with its canonical defining cocycle consisting of all simple open subsets of X and the canonical projections onto the corresponding quotient spaces of plaques. Foliated spaces with boundary or corners can be defined similarly. Indeed, we will only need the connectedness and local path connectedness of the fibers of the submersions Pi. Many interesting examples of foliated spaces are given in [5]. For a defining cocycle {Ui,Pi} of T, the homeomorphisms hfj, given by its compatibility condition, generate a pseudogroup Ti acting on T = |_|i Tj, and {pi} generates a morphism V : X —> Ti. Let {U^,p'a} be another defining cocycle of T with p'a :£/„—> T'a, which induces a pseudogroup Ti' acting on T = \JaTa a n d a morphism V' : X -> Ti'. Then {UhPi} U {U'a,p'a} induces a pseudogroup Ti" acting on T" = T U f ' , defining a canonical isomorphism $ : Ti —> Ti' so that $ o V = V1. The "transverse dynamics" or "transverse structure" of T is described by the equivalence class of Ti, which is called holonomy pseudogroup. It has a canonical representative induced by the canonical defining cocycle, which is denoted by Hol(jF) and will be called holonomy pseudogroup too. Let T and Q be foliated structures on spaces X and Y. Their product is the foliated structure J^xQ onXxY whose leaves are the products of leaves of T and Q. Observe that Hol(.F x Q) is equivalent to Hol(.F) x Hol(£). 4
Holonomy morphisms of foliated maps
Let X and Y be foliated spaces with foliated structures T and Q. A foliated map f : T —> Q is a map / : X —> Y which maps leaves of T to leaves of Q; it induces a map XjT —• Y/Q. The set of continuous foliated maps T —> £ is denoted by C(.F,(7); the notation C°°(.F,£) is used for the set of C°°
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
5
foliated maps when T and Q are C°° foliations. Continuous foliated maps, with the operation of composition, form a category denoted by Fol. The concept of foliated map can be similarly defined for singular foliations. Let {Ui,pi} and {V^,^} be defining cocycles of T and Q with pt : Ui —> Ti and p'a : Va —> T'a, which induce pseudogroups H and H', and morphisms P : X ^ H and V' : Y ^ H'. Given any / G C(T, G), we can choose {Ui,pi} and {Va,p'a} such that / maps each fiber of pi to a fiber of p ^ for some mapping i >-> a*. So there are continuous maps Q a i satisfying 4>i°pi= p'a. of\Ut. Then {, ip £ is complete. (ii) $ generates a morphism $ :H —*H'. (iii) The maps in $ are C°° along the leaves of the singular foliation defined by the orbit closures, with continuous leafwise derivatives of arbitrary order. In Theorem 9.1, the morphism $ is called the closure of $. The continuity of leafwise derivatives, used in (iii), makes sense even for singular foliations!
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Properties (i) and (ii) of Theorem 9.1 can be reduced to the following result. Proposition 9.2 For all 4> G $ and all x G dorrufi, there is an open neighborhood U of x in dome/) satisfying the following properties (i) There is a (compact-open) neighborhood O of idu in Hu such that, for all h G O, we have h(U) C domefr, and there is some h' G Ti' with <j>{U) C domh' and h' o = <j>oh on U. (ii) For all h[,h2 G TV with <j>(U) C domh'x n domh'2, if h[ o = h'2 o on some neighborhood of x, then h[ o = h'2 ° 4> on U, Let us show that Proposition 9.2 implies (i) and (ii) of Theorem 9.1. To prove that $ generates $, it is enough to prove the condition (hi') of generation. Take <j>, tp G $ , h G H and x G dom(f>r\dom(il>oh). Consider the open neighborhood U of x given by Proposition 9.2. We can also assume that (U, U) is a completeness pair for Ti by taking U small enough. Then there is a sequence of maps hn G Ti, defined on some open neighborhood of x in U, so that hn —> h with respect to the compact-open topology. On the one hand, the sequence gn = h~l o h G H is defined in some fixed neighborhood of x, where it converges to the identity. Since (U, U) is a completeness pair, there is a sequence gn G Tlu so that 7(
