# Books on foliations

I am looking for resources (books, notes, lecture video, etc. anything will do although printed material in English is preferable) on foliations which satisfy some or all of the following constraints.

• Prerequisites: I am familiar with algebraic topology (in the geometric style, as in Hatcher), differential topology (as in Guillemin-Pollack), and Riemannian geometry (as in do Carmo) along with all other standard undergraduate topics (by which I mean content covered in standard textbooks in real and complex analysis, linear and basic algebra, commutative algebra, classical algebraic geometry, point set topology, curves and surfaces). I am also familiar with characteristic classes (Morita's differential forms book and Madsen-Tornehave), basic symplectic geometry (da Silva), and basic topological/measure theoretic dynamics (earlier chapters of Brin-Stuck).

I am also familiar with physics (general relativity using Caroll, classical mechanics using Goldstein, etc.) if it helps.

Ideally, I am looking for two types of resources. (Notice that the two are not mutually exclusive.)

• An exposition of the theory which has a strong geometric taste (much like Hatcher's books) ideally with a lot of pictures and concrete examples. Ideally, the book connects new ideas introduced in the book with older ideas (described in "prerequisites" above).

• Collection of problems which allows one to practice applying the theory. I prefer exercises which are not just filling in technical details which the author did not have time for. Instead, I prefer something which allows one to a.) learn key heuristics, and ideally b.) get a sense on why the theory will be important later in one's studies.

So far, I have the following books:

• Tamura, Topology of Foliations: An Introduction
• Calegari, Foliations and the Geometry of 3–Manifolds
• I see in Wikipedia these references (neither of which I know; just references to me): Candel, Alberto; Conlon, Lawrence (2000). Foliations I. Graduate Studies in Mathematics. 23. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0809-5. Candel, Alberto; Conlon, Lawrence (2003). Foliations II. Graduate Studies in Mathematics. 60. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0809-5. May 22 '20 at 21:30
• There are the two volumes of G. Hector and U. Hirsch, Introduction to the geometry of foliations. (Vieweg, Braunschweig, 1981) which are fairly geometric and explicit. In the first volume, they discuss the classification of foliations on surfaces in some detail. May 22 '20 at 22:22

Geometric Theory of Foliations by César Camacho and Alcides Lins Neto in Portuguese, or in English thanks to Sue Goodman's fantastic translation.

I think it does almost everything you're asking for in terms of pictures, examples, and lovely exercises beyond just letting readers fill in details, though some theorems do leave proofs to the reader.

As is pointed out in the comments the two books by Candel and Conlon and your inclusion in your post of Calegari's book are amazing, but for rather different reasons.

Candel and Conlon could be said to provide the opposite of a geometric experience, in the sense that it provides an opportunity to learn the theory by manipulating the symbols of differential geometry's formalism. The degree of precision and generality is probably unrivaled.

Calegari paints a quite different picture, giving a sophisticated and contextualized presentation. His choice of material to collect in that book seems to have been prescient, in that the book was completed 5 years prior to the publication of, what is now known as, the L-space conjecture in ~2012. The content in Calegari's book is a fantastic preparation for understanding the state of the art techniques in low dimensional foliation theory to attack the L-space conjecture. Worth mentioning is that a proof of the L-space conjecture would provide us with a new, Ricci free, proof of the Poincaré conjecture.

• Could you elaborate on your last sentence? Surely it’s probable that any solution to the L-space conjecture will rely on the Poincaré conjecture, or even Geometrisation?
– HJRW
Jun 6 '20 at 10:24
• The proof in the graph manifold case does not depend on Poincaré or Geometrisation. To be too brief with Boyer and Clay's work, they use a gluing result for horizontal foliations in Seifert fibered spaces, and when that doesn't quite work, it's a gluing result for contact approximations of foliations. The full graph manifold case is a combination of Boyer-Clay and Sarah Rasmussen. András Juhász writes about a route to the Poincaré through the L-space conjecture in arxiv.org/abs/1310.3418. Which to be honest does depend on an additional conjecture, but sort of only open in hyperbolic. Jun 6 '20 at 18:59
• Graph manifolds are geometrisable by definition! Juhasz’s paper looks interesting — thanks!
– HJRW
Jun 7 '20 at 6:48
• Haha yes of course they are geometrisable. I see now what you mean, if someone deals with hyperbolic manifolds and then follows the same thread through a gluing result using geometrization tori then yea, it seems eminently reasonable it will depend on Perelman. The Ricci free stuff I was referring to is in the direction of Thurston's universal circles. Thurston's slithering preprints and following work to prove geometrisation by taut foliations. The state of the art efforts I mentioned being lifting universal circle representations to actions on $\mathbb{R}$ to certify left orderability. Jun 7 '20 at 7:51