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The title of this post is a nod to the excellent and well-known Div, grad, curl and all that by Harry Schey (and perhaps also to the lesser-known sequel to one of the more consoling histories of Great Britain), and the purpose is to explain how to generalize these differential operators (familiar to electrical engineers and undergraduates taking vector calculus) and a few other ones from Euclidean 3-space to arbitrary Riemannian manifolds. I have a complicated relationship with the subject of Riemannian geometry; when I reviewed Dominic Joyce’s book Riemannian holonomy groups and calibrated geometry for SIAM reviews a few years ago, I began my review with the following sentence:
Riemannian manifolds are not primitive mathematical objects, like numbers, or functions, or graphs. They represent a compromise between local Euclidean geometry and global smooth topology, and another sort of compromise between precognitive geometric intuition and precise mathematical formalism.
Don’t ask me precisely what I meant by that; rather observe the repeated use of the key word compromise. The study of Riemannian geometry is — at least to me — fraught with compromise, a compromise which begins with language and notation. On the one hand, one would like a language and a formalism which treats Riemannian manifolds on their own terms, without introducing superfluous extra structure, and in which the fundamental objects and their properties are highlighted; on the other hand, in order to actually compute or to use the all-important tools of vector calculus and analysis one must introduce coordinates, indices, and cryptic notation which trips up beginners and experts alike.
For almost 50 years, Paul Sally was a towering figure in mathematics education at the University of Chicago. Although he was 80 years old, and had two prosthetic legs and an eyepatch (associated with the Type 1 diabetes he had his whole life), it was nevertheless a complete shock to our department when he passed away last December, and we struggled just to cover his undergraduate teaching load this winter and spring. As my contribution, I have been teaching an upper-division undergraduate class on “topics in geometry”, which I have appropriated and repurposed as an introduction to the classical geometry and topology of surfaces.
I have tried to include at least one problem in each homework assignment which builds a connection between classical geometry and some other part of mathematics, frequently elementary number theory. For last week’s assignment I thought I would include a problem on the well-known connection between Pythagorean triples and the modular group, perhaps touching on the Euclidean algorithm, continued fractions, etc. But I have introduced the hyperbolic plane in my class mainly in the hyperboloid model, in order to stress an analogy with spherical geometry, and in order to make it easy to derive the identities for hyperbolic triangles (i.e. hyperbolic laws of sines and cosines) from linear algebra, so it made sense to try to set up the problem in the language of the orthogonal group , and the subgroup preserving the integral lattice in .
When I started in graduate school, I was very interested in 3-manifolds, especially Thurston’s geometrization conjecture. Somehow in dimension 3, there is a marvelous marriage of flexibility and rigidity: generic 3-manifolds are flexible enough to admit hyperbolic structures — i.e. Riemannian metrics of constant curvature -1, modeled on hyperbolic space. But these structures are so rigid that they are determined up to isometry (!) entirely by the fundamental group of the manifold, and provide a bridge from topology to the rigid world of number fields and arithmetic. 3-manifolds, especially the hyperbolic ones, display an astonishing range of interesting phenomena, so that even though the individual manifolds are discrete and rigid, they come in infinite families parameterized by Dehn surgery. When Perelman proved Thurston’s conjecture, I gradually moved away from 3-manifold topology into some neighboring fields such as dynamics and geometric group theory; subjectively this move felt to me like a transition from a baroque world of highly intricate, finely tuned and beautiful objects to more rough and disordered domains in which the rule was chaos and disorder, and where one had to restrict attention and focus to find the kinds of structured objects that one can say something about mathematically. In these new domains my familiarity with 3-manifold topology was always extremely useful to me, but almost always as a source of inspiration or analogy or example, rather than that some specific theorem about 3-manifolds could be used to say something about groups in general, or dynamical systems in general, or whatever. Many important recent developments in geometric group theory are generalizations of geometric ideas which were first identified or studied in the world of 3-manifolds; but there was not much connection at the deepest level, at least as far as I could see.
This impression was dramatically shaken by Agol’s proof of the virtual Haken conjecture and virtual fibration conjectures in 3-manifold topology by an argument which depends for one of its key ingredients on the theory of non-positively curved cube complexes — a subject in geometric and combinatorial group theory which, while inspired by key examples in low-dimensions (especially surfaces in the hands of Scott, and graphs in the hands of Stallings), is definitely a high-dimensional theory with no obvious relations to manifolds at all. Even so, the transfer of information in this case is still from the “broad” world of group theory to the “special” world of 3-manifolds. It shows that 3-manifold topology is even richer than hitherto suspected, but it does not contradict the idea that the beautiful edifice of 3-manifold topology is an exceptional corner in the vast unstructured world of geometry.
I have just posted a paper to the arXiv, coauthored with Henry Wilton, and building on prior work I did with Alden Walker, that aims to challenge this idea. Let me quote the first couple of paragraphs of the introduction:
Geometric group theory was born in low-dimensional topology, in the collective visions of Klein, Poincaré and Dehn. Stallings used key ideas from 3-manifold topology (Dehn’s lemma, the sphere theorem) to prove theorems about free groups, and as a model for how to think about groups geometrically in general. The pillars of modern geometric group theory — (relatively) hyperbolic groups and hyperbolic Dehn filling, NPC cube complexes and their relations to LERF, the theory of JSJ splittings of groups and the structure of limit groups — all have their origins in the geometric and topological theory of 2- and 3-manifolds.
Despite these substantial and deep connections, the role of 3-manifolds in the larger world of group theory has been mainly to serve as a source of examples — of specific groups, and of rich and important phenomena and structure. Surfaces (especially Riemann surfaces) arise naturally throughout all of mathematics (and throughout science more generally), and are as ubiquitous as the complex numbers. But the conventional view is surely that 3-manifolds per se do not spontaneously arise in other areas of geometry (or mathematics more broadly) amongst the generic objects of study. We challenge this conventional view: 3-manifolds are everywhere.
It’s been a while since I last blogged; the reason, of course, is that I felt that I couldn’t post anything new before completing my series of posts on Kähler groups; but I wasn’t quite ready to write my last post, because I wanted to get to the bottom of a few analytic details in the notorious Gromov-Schoen paper. I am not quite at the bottom yet, but maybe closer than I was; but I’m still pretty far from having collected my thoughts to the point where I can do them justice in a post. So I’ve finally decided to put Kähler groups on the back burner for now, and resume my usual very sporadic blogging habits.
So the purpose of this blog post is to advertise that I wrote a little piece of software called kleinian which uses the GLUT tools to visualize Kleinian groups (or, more accurately, interesting hyperbolic polyhedra invariant under such groups). The software can be downloaded from my github repository at
and then compiled from the command line with “make”. It should work out of the box on OS X; Alden Walker tells me he has successfully gotten it to compile on (Ubuntu) Linux, which required tinkering with the makefile a bit, and installing freeglut3-dev. There is a manual on the github page with a detailed description of file formats and so on.
In this post I hope to start talking in a bit more depth about the global geometry of compact Kähler manifolds and their covers. Basic references for much of this post are the book Fundamental groups of compact Kähler manifolds by Amoros-Burger-Corlette-Kotschick-Toledo, and the paper Kähler hyperbolicity and L2 Hodge theory by Gromov. It turns out that there is a basic distinction in the world of compact Kähler manifolds between those that admit a holomorphic surjection with connected fibers to a compact Riemann surface of genus at least 2, and those that don’t. The existence or non-existence of such a fibration turns out to depend only on the fundamental group of the manifold, and in fact only on the algebraic structure of the cup product on ; thus one talks about fibered or nonfibered Kähler groups.
If X is a connected CW complex, by successively attaching cells of dimension 3 and higher to X we may obtain a CW complex Y for which the inclusion of X into Y induces an isomorphism on fundamental groups, while the universal cover of Y is contractible (i.e. Y is a with the fundamental group of X). The (co)-homology of Y is (by definition) the group (co)-homology of the fundamental group of X. Since Y is obtained from X by attaching cells of dimension at least 3, the map induced by inclusion is an isomorphism in dimension 0 and 1, and an injection in dimension 2 (dually, the map is a surjection, whose kernel is the image of under the Hurewicz map; so the cokernel of measures the pairing of the 2-dimensional cohomology of X with essential 2-spheres).
A surjective map f from a space X to a space S with connected fibers is surjective on fundamental groups. This basically follows from the long exact sequence in homotopy groups for a fibration; more prosaically, first note that 1-manifolds in S can be lifted locally to 1-manifolds in X, then distinct lifts of endpoints of small segments can be connected in their fibers in X. A surjection on fundamental groups induces an injection on in the other direction, and by naturality of cup product, if is a subspace of on which the cup product vanishes identically — i.e. if it is isotropic — then is also isotropic. If S is a closed oriented surface of genus g then cup product makes into a symplectic vector space of (real) dimension 2g, and any Lagrangian subspace V is isotropic of dimension g. Thus: a surjective map with connected fibers from a space X to a closed Riemann surface S of genus at least 2 gives rise to an isotropic subspace of of dimension at least 2.
So in a nutshell: the purpose of this blog post is to explain how the existence of isotropic subspaces in 1-dimensional cohomology of Kähler manifolds imposes very strong geometric constraints. This is true for “ordinary” cohomology on compact manifolds, and also for more exotic (i.e. ) cohomology on noncompact covers.
One of the nice things about living in Hyde Park is the proximity to the University of Chicago. Consequently, over the summer I came in to the department from time to time to work in my office, where I have all my math books, fast internet connection, etc. One day in early September (note: the Chicago quarter doesn’t start until October, so technically this was still “summer”) I happened to run in to Volodya Drinfeld in the hall, and he asked me what I knew about fundamental groups of (complex) projective varieties. I answered that I knew very little, but that what I did know (by hearsay) was that the most significant known restrictions on fundamental groups of projective varieties arise simply from the fact that such manifolds admit a Kähler structure, and that as far as anyone knows, the class of fundamental groups of projective varieties, and of Kähler manifolds, is the same.
Based on this brief interaction, Volodya asked me to give a talk on the subject in the Geometric Langlands seminar. On the face of it, this was a ridiculous request, in a department that contains Kevin Corlette and Madhav Nori, both of whom are world experts on the subject of fundamental groups of Kähler manifolds. But I agreed to the request, on the basis that I (at least!) would get a lot out of preparing for the talk, even if nobody else did.
Anyway, I ended up giving two talks for a total of about 5 hours in the seminar in successive weeks, and in the course of preparing for these talks I learned a lot about fundamental groups of Kähler manifolds. Most of the standard accounts of this material are aimed at people whose background is quite far from mine; so I thought it would be useful to describe, in a leisurely fashion, and in terms that I find more comfortable, some elements of this theory over the course of a few blog posts, starting with this one.
A couple of weeks ago, my student Yan Mary He presented a nice proof of Liouville’s theorem to me during our weekly meeting. The proof was the one from Benedetti-Petronio’s Lectures on Hyperbolic Geometry, which in my book gets lots of points for giving careful and complete details, and being self-contained and therefore accessible to beginning graduate students. Liouville’s Theorem is the fact that a conformal map between open subsets of Euclidean space of dimension at least 3 are Mobius transformations — i.e. they look locally like the restriction of a composition of Euclidean similarities and inversions on round spheres. This implies that the image of a piece of a plane or round sphere is a piece of a plane or round sphere, a highly rigid constraint. This sort of rigidity is in stark contrast to the case of conformal maps in dimension 2: any holomorphic (or antiholomorphic) map between open regions in the complex plane is a conformal map (and conversely). The proof given in Benedetti-Petronio is certainly clear and readable, and gives all the details; but Mary and I were a bit unsatisfied that it did not really provide any geometric insight into the meaning of the theorem. So the purpose of this blog post is to give a short sketch of a proof of Liouville’s theorem which is more geometric, and perhaps easier to remember.
Yesterday and today Marty Scharlemann gave two talks on the Schoenflies Conjecture, one of the great open problems in low dimensional topology. These talks were very clear and inspiring, and I thought it would be useful to summarize what Marty said in a blog post, just for my own benefit.
The story starts with the following classical theorem, usually called the Jordan curve theorem, or Jordan-Schoenflies theorem:
Theorem (Jordan-Schoenflies): Let P be a simple closed curve in the plane. Then its complement has a unique bounded component, whose closure is homeomorphic to the disk in such a way that P becomes the boundary of the disk.
In order to make the relationship between the two complementary components more symmetric, one could express this theorem by saying that a simple closed curve P in the 2-sphere separates the 2-sphere into two components X and Y, each of which has closure homeomorphic to a disk with P as the boundary.
Based on this simple but powerful fact in dimension 2, Schoenflies asked: is it true for every n that every n-sphere P in the (n+1)-sphere splits the (n+1)-sphere into two standard (n+1)-balls?