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This book contains the lecture notes of four courses on several topics with rather different flavor, which are linked by their relation with Foliation Theory. These courses were given at the Centre de Recerca Matemàtica (CRM), Bellaterra (Barcelona), in 2010. They show the diversity of ideas converging around Foliation Theory. Besides giving a general overview, the courses will be very helpful for any reader that wants to get quickly introduced to any of these lines of research.
The first one, by Masayuki Asaoka, is about the connection between deformation of locally free actions of Lie groups with leafwise cohomology. Here, the locally free actions define (regular) foliations whose leaves are the connected components of the orbits, and then their leafwise cohomology is defined. It a version of the de Rham cohomology for foliations, taking differential forms on the leaves that are smooth on the ambient manifold. It begins by introducing the closed subspace \(\mathcal A(M,G)\subset C^\infty(M\times G,M)\) of (right) actions of a Lie group \(G\) on a manifold \(M\), with the \(C^\infty\) topology, and its closed subspace \(\mathcal A_{LF}(M,G)\) of locally free actions (the isotropy groups are discrete). Then the basic concept of (\(C^\infty\)) conjugation between \(G\)-actions means that they correspond by a diffeomorphism up to some automorphism of \(G\). The closed subspace \(\mathcal A_{LF}(\mathcal F,G)\subset\mathcal A_{LF}(M,G)\) is defined by requiring that the orbit foliation is \(\mathcal F\). Two actions in \(\mathcal A_{LF}(\mathcal F,G)\) are called (\(C^\infty\)) parameter equivalent if they are conjugated by a diffeomorphism that preserves each leaf. This gives rise to the concept of (\(C^\infty\)) parameter rigidity of an action \(\rho_0\in\mathcal A_{LF}(\mathcal F,G)\), meaning that any \(\rho\in\mathcal A_{LF}(\mathcal F,G)\) is parameter equivalent to \(\rho_0\), and its local version, which only requires this condition for all \(\rho\) in some neighborhood of \(\rho_0\) in \(\mathcal A_{LF}(\mathcal F,G)\). Several examples are given illustrating these concepts. For instance, every action of a compact group on a closed manifold is rigid (Palais). Then the author defines a \(C^\infty\) family in \(\mathcal A_{LF}(M,G)\) parametrized by a manifold \(\Delta\); the space of such families is denoted by \(\mathcal A_{LF}(M,G;\Delta)\). If \(\Delta\) is an open neighborhood of \(0\) in a finite-dimensional vector space, then a family \((\rho_\mu)_{\mu\in\Delta}\in\mathcal A_{LF}(M,G;\Delta)\) is called a (finite-dimensional) deformation of \(\rho=\rho_0\). This deformation is called locally complete if any action in some neighborhood of \(\rho\) in \(\mathcal A_{LF}(M,G)\) is conjugate to some \(\rho_\mu\), and it is called locally transverse if any family in \(\mathcal A_{LF}(M,G;\Delta)\) sufficiently close to \((\rho_\mu)_{\mu\in\Delta}\) contains an action conjugate to \(\rho\). As above, we also have the foliation version of these concepts, being locally complete or locally transverse in \(\mathcal A_{LF}(\mathcal F,G;\Delta)\), where parameter equivalence is used instead of just conjugation. These concepts are illustrated for flows (\(G=\mathbb R\)). For instance, Katok’s conjecture is explained, stating that every parameter rigid flow on a closed manifold is conjugate to a Diophantine flow on a torus. A solution in dimension \(3\) was independently given by Forni, Kocsard and Matsumoto.
Then the leafwise cohomology is defined, with special emphasis on methods of computation by Fourier analysis and Mayer-Vietoris arguments. Examples of computation are given, like in the case of a Diophantine linear flow on a torus or the Weyl chamber flow. The relation between parameter rigidity and leafwise cohomology is explained next for some types of actions, giving results of Matsumoto and Mitsumatsu, Dos Santos, Maruhashi and the author. Then Hamilton’s criterium for local rigidity is explained, and several applications are given (by El Kacimi Alaoui and Nicolau, and Matsumoto and Mitsumatsu). Other methods to study local rigidity and leafwise cohomology are also explained, giving results by Katok and Spatzier (for the Weyl chamber flow), Moser, Mieczkowski, Damjanovik and Katok, Ghys, and the author.
The second course, by Aziz El Kacimi-Alaoui, is an introduction to Foliation Theory, with special emphasis on examples and transverse geometric structures. It begins by giving basic concepts and constructions: foliation, leaf, foliated cocycle, pull-back foliation, conjugation, Frobenius theorem and holonomy. Holonomy transformations are defined by moving local transversals along the leaves; this is the transverse dynamics of the foliation, which plays a key role. Next, the concept of transverse structure is introduced, and some examples of such structures are given, which define classes of foliations: Lie foliations, transversely parallelizable foliations, Riemannian foliations, transversely homogeneous foliations, and transversely holomorphic foliations. Some results about specific structures are recalled, like Molino’s description of Riemannian foliations. All of this is illustrated with many concrete examples. The theory of codimension one foliations is recalled next, introducing basic concepts and stating some key theorems that illustrate its wealth and subtleties. Then the course is oriented to topics where the author has made relevant contributions. There is a digression about basic Global Analysis, studying index theory for invariant transversely elliptic differential operators. For instance, the basic de Rham complex of a foliation on a closed manifold satisfies some analytic properties similar to the de Rham complex on a closed manifold: Hodge decomposition and finite dimension, giving rise to a basic Euler characteristic. The same is explained also for the basic Dolbeault complex. Another important point of view is the study of leafwise elliptic differential operators or complexes, which can be made by keeping smoothness on the ambient manifold, like the leafwise de Rham complex giving the leafwise cohomology, or in the sense of Connes, in the context of non-commutative geometry. This course is finished with some open problems: a basic Atiyah-Singer index theorem, existence of transversely elliptic operators, homotopy invariance of the basic cohomology, existence of complex foliations of codimension one, the \(\bar\partial\) version of the leafwise cohomology for holomorphic foliations, and deformations of Lie foliations.
The third course, by Steven Hurder, is devoted to the study of the dynamics of foliations. Many concepts from topological dynamics are extended to the setting of foliations via holonomy transformations, where new perspectives are achieved. For instance, transverse invariant measures are defined, and there is a classical result about their existence (Sacksteder). Other typical dynamical concepts that are extended to foliations are minimal sets, equicontinuity, distality and proximality. Looking at the derivatives of holonomy maps for “long time”, the author explains a partition of the foliated manifold into three sets, consisting of elliptic, parabolic and hyperbolic points. Some results relating these sets with invariant measures are proved in the notes. Another dynamical concept extended to foliations is the growth type of the orbits or leaves, which is illustrated with examples. A more involved extended concept is the geometric entropy, introduced by Ghys, Langevin and Walczak, and also studied by the author. Related concepts are the expansion growth and the local geometric entropy. Several results concerning these concepts are proved, and they are studied for the so called Hirsch foliations. Moreover open problems are proposed about them, and several theorems are recalled, relating the entropy with invariant measures, distality, the Godbillon-Vey class, resilient leaves, hyperbolic points, and the so called exponential quivers. Next minimal sets are studied, with special emphasis on their relation with the elliptic-parabolic-hyperbolic partition. Another section raises the classification problem of foliations up to conjugacy, cobordism or other natural equivalence relations. A few results in this direction are explained, and mainly open problems are proposed. Then the concept of matchbox manifold is explained (a compact metrizable space with a “foliation” whose local transversals are Cantor sets); these objects play a role in Foliation Theory as candidates for minimal sets. For instance, classical solenoids are matchbox manifolds, as well as their direct generalization using compact coverings of closed manifolds. If the coverings are normal, we get the so called McCord solenoids. A theorem of the author with Clark is explained, showing that any equicontinuous matchbox manifold is a solenoid, and any homogeneous matchbox manifold is a McCord solenoid. The course is finished with the application of shape theory to dynamics, and its extension to foliations.
The last course, by Ken Richardson, is about a solution to the basic index problem for Riemannian foliations, as already explained in the course by Aziz El Kacimi-Alaoui. Via Molino’s description of these foliations, the original index problem is transformed into a problem about the equivariant index for actions of compact Lie groups on closed manifolds, as originally studied by Atiyah. In turn, this also fits into the more general setting of singular Riemannian foliations. Much was known about such equivariant index theory, except a precise general index formula (which was only known in particular cases). The author has achieved such a formula in collaboration with Brüning and Kamber. His course explains all concepts needed to understand it: Dirac operators, Atiyah-Singer index theorem, basic Dirac operators, basic de Rham operator, natural examples, equivariant index theory for \(G\)-manifolds, isotropy groups, stratification of \(G\)-manifolds, equivariant desingularization, decomposition of equivariant bundles, and canonical isotropy of \(G\)-bundles. Then their equivariant index formula is stated, yielding a basic index formula for Riemannian foliations. The formula is obtained by tracking back the changes of the index through the desingularization procedure. One of the main tools is the invariance of the index by deformations. Besides this theory, there is a long list of proposed exercises, many of them considering concrete examples.