TITLE: What is a foliation and why should I care?
QUESTION [62 upvotes]: The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential geometry course (I was the Reeb foliation) and I didin't pay many attention to it. In the meanwhile I get interested in the noncommutative theory in particular in $C^*$-algebras. While reading about Noncommutative Geometry I came across foliations as the one of the main motivating examples of the theory. I learned that in general the space of leaves of the foliation is badly behaved as a topological space and I believe that it is more worthwile to deal with these spaces using algebraic methods. But I don't have something like a mental picture of what foliations is and why should I even care about those objects?

REPLY [7 votes]: Here's why I care about foliations. It is always interesting when a structure can be expressed in terms of simpler structures. For instance a torus is the union of circles making it into a cartesian product. Or a Klein bottle is the union of circles making it into a non-trivial fibre bundle.
A foliation is a description of an n-manifold M as the union of submanifolds of fixed lower dimension k, whose tangent planes fit together cleanly. (I.e., such that every point of M has a neighborhood that is decomposed topologically just as n-dimensional Euclidean space is the cartesian product of k-dimensional Euclidean space and (n-k)-dimensional Euclidean space.) This is a natural generalization of two things: a) the concept of a fibre bundle, and b) the decomposition of a manifold into the trajectories of a non-singular vector field.
A foliation can be defined in terms of the reduction of a manifold's atlas to a certain simple pseudogroup.
The quintessential example of a foliation is the Reeb foliation of the 3-sphere. Although the 3-sphere possesses infinitely differentiable codimension-1 foliations such as the Reeb foliation, it is a beautifully subtle fact, proved by André Haefliger, that it has no real-analytic one. Another fascinating fact, proved by Sergei Novikov, is that every codimension-1 foliation of the 3-sphere must possess a compact leaf (which is in fact a torus). Yet another fascinating fact, proved by Krystyna Kuperberg, is that the 3-sphere possesses a real-analytic 1-foliation that has no closed trajectory. (This counterexample was discovered only 43 years after the original conjecture, by Herbert Seifert, suggested that every 1-foliation of the 3-sphere must have a closed trajectory.)