4-spheres from fibered knots

I was at UC Riverside this past weekend, attending the regional meeting, and giving a talk in a special session on knot theory in memory of the late Xiao-Song Lin. After lunch, I joined in a conversation between Rob Kirby and Mike Freedman on the recent flurry of activity this summer, in which Selman Akbulut and Bob Gompf independently showed that the Cappell-Shaneson manifolds — smooth -manifolds known since Freedman’s work to be homeomorphic to — are in fact diffeomorphic to the standard smooth (actually, Cappell-Shaneson’s manifolds have the additional feature that they admit a free action, giving rise to fake ’s, which was actually their original interest).

Apparently these constructions had somewhat altered the experts’s (i.e. Freedman and Kirby) feelings about whether the smooth -dimensional Poincaré conjecture is likely to be true. The Cappell-Shaneson manifolds are constructed by doing surgery on certain torus bundles over a circle — those with monodromy chosen so that the resulting torus bundles have the homology of a . A suitable surgery, killing the factor makes the manifold into homology ’s, and also kills the subgroup of the fundamental group normally generated by the factor. On the other hand, everything else in the fundamental group “comes from” the torus, whose fundamental group is abelian, and therefore the resulting manifold is simply-connected. Since it is a homology -sphere, is it therefore a homotopy -sphere, and consequently (by Freedman), a topological -sphere.

Gompf shows these -spheres are standard by showing that a certain move — which simplifies the monodromy of the fiber — can be realized by a diffeomorphism. The move is an example of what is known to -manifold topologists as a “log transform” (and to -manifold topologists as “Dehn surgery times ”). A log transform takes as input a smooth embedded torus. A tubular neighborhood of this torus is a product whose boundary is a -torus . This tubular neighborhood is removed, and reglued by an automorphism of the factor. Usually a log transform will change the topology of the manifold, or at least the smooth structure. But in this case, the surgered torus is contained in a fiber, and the log transform can be shown to be isotopic to the identity, by using the monodromy of the fibration (technically, the monodromy of the fibration produces a once-punctured torus in the bundle with boundary on the curve along which the log transform “twists”, but after doing surgery to produce the homology ’s, this once-punctured torus is “capped” to become a smooth disk).

The point of this blog post is to show how to construct many, many other smooth -manifolds which are topological -spheres, and for which Gompf’s method of showing they are standard does not work. Are these manifolds counterexamples to the smooth -dimensional Poincaré conjecture? I am really not the person to ask.

The construction takes as input a fibered knot — i.e. a knot in the -sphere whose complement fibers over a circle. In other words, there is a fibration , where is a (minimal genus) Seifert surface for the knot . The fibration of spaces gives rise to a short exact sequence of fundamental groups (in general, one gets a long exact sequence of homotopy groups, but the spaces are all ’s — i.e. their homotopy groups in dimension other than all vanish). Since has boundary, the fundamental group of is free and finitely generated of rank , where is the genus. The fundamental group of is . So one exhibits as an HNN extension of a free group, where the meridian acts by conjugation on the free group by some automorphism .

Since is a knot in , the homology of is equal to in dimension . Moreover, since putting back in recovers , it follows that the fundamental group is normally generated by the meridian (which also generates the in ). For the moment everything is -dimensional, but there is a trick to promote this to dimensions. In place of the surface , consider the -manifold . In other words, is obtained by doubling a handlebody of genus . The fundamental group is free of rank . Now one builds a bundle over with monodromy ; call this -manifold . The existence of such a manifold depends on being able to realize any automorphism of a free group by a homeomorphism of a doubled handlebody; one way to see this is to observe that is generated by Nielsen moves — interchanging generators, replacing generators by their inverses, and replacing generators by . These moves are all realizable by homeomorphisms of doubled handlebodies, the last by a “handle slide”.

Now, observe that , and is normally generated by a loop representing the circle direction. Moreover, . To compute , observe that by Poincaré duality. If the action of on (a free abelian group of rank ) is represented by a matrix , then the action on is represented by the transpose . The fact that is equivalent to the fact that the first homology of dies in the bundle; i.e. ; hence , and for the same reason, the second homology of dies in the bundle, and . By (-dimensional) Poincaré duality, , and we see that is a homology .

A tubular neighborhood of the loop is a product , since is orientable. The boundary of this is . So we drill out and glue in a product to produce . Drilling out does not affect the fundamental group, by Seifert van-Kampen, and the fact that the inclusion is an isomorphism on . On the other hand, filling in a has the effect of killing the meridian , and therefore (by the discussion above), killing completely; i.e. is simply-connected. Hence . Drilling out a circle and gluing back a sphere increases Euler characteristic by ; since the rank of and have both gone down by , it follows that the rank of is still , and since the fundamental group is trivial, . So is a smooth, simply-connected homology sphere, which is to say, a smooth -manifold which is topologically .

Back in June, Freedman-Gompf-Morrison-Walker described a way to use Rasmussen’s -invariant to detect exotic ’s, and proposed trying this invariant out on the Cappell-Shaneson examples (see Scott Morrison’s post about that here). Is it feasible to compute the invariants on these new examples?

(Update:) Mike Freedman pointed out in an email that these spheres are all standard, and give rise to interesting open-book decompositions of with binding a knot and fiber where is the fiber of the fibration of . This is related to the familiar fact that -twisted spun knots are unknotted (see e.g. Rolfsen, p. 339) — an interesting exercise in visualization. (Updated update:) there are some interesting variations on this construction which give other ’s that are not obviously standard (at least not in the same obvious way). If I can work out some good examples, I’ll try to describe them.

Minimal laminations with leaves of different conformal types

The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it comes from.

The example comes from the idea of a Riemann surface lamination. This is an object that geometrizes some ideas in 1-dimensional complex analysis. The basic idea is simple: given a noncompact infinite Riemannian -manifold , one gives it a new topology by declaring that two points on the surface are “close” in the new topology if there are balls of big radius in the surface centered at the two points which are “almost isometric”. Points that were close in the old topology are close in the new topology, but points that might have been far away in the old topology can become close in the new. For example, if is a covering space of some other Riemannian surface , then points in the orbit of the deck group are “infinitely close” in the new topology. This means that the resulting topological space is not Hausdorff; one “Hausdorffifies” by identifying pairs of points that are not contained in disjoint open sets, and the quotient recovers the surface (assuming that the metric on is sufficiently generic; otherwise, it recovers modulo its group of isometries). Morally what one is doing is mapping into the space of pointed locally compact metric spaces (which is itself a locally compact topological space), and giving it the subspace topology. In more detail, a point in is a pair where is a locally compact metric space, and is a point. A sequence converges to if there are metric balls around of diameter going to infinity, metric balls around also of diameter going to infinity, and isometric inclusions of into metric spaces in such a way that the Hausdorff distance between the images of and in goes to zero as . Any locally compact metric space has a tautological map to , where each point is sent to the point . Gromov showed (see section 6 of this paper) that the space itself is locally compact; in fact, this follows in an obvious way from the Arzela-Ascoli theorem.

If has bounded geometry — i.e. if the injectivity radius is uniformly bounded below, and the curvature is bounded above and below — then the image of in is precompact, and its closure is a compact metric space . The path components of are exactly the Riemann surfaces which are arbitrarily well approximated (in the metric sense) on every compact subset by compact subsets of . If you were wandering around on such a component , and you wandered over a compact region, and were only able to measure the geometry up to some (arbitrarily fine) definite precision, you could never rule out the possibility that you were actually wandering around on . Topologically, is a Riemann surface lamination; i.e. a locally compact topological space covered by open charts of the form where is an open two-dimensional disk, where is totally disconnected, and where the transition between charts preserves the decomposition into pieces , and is smooth (in fact, preserves the Riemann surface structure) on the slices, in the overlaps. The unions of “surface” slices — i.e. the path components of — piece together to make the leaves of the lamination, which are (complete) Riemann surfaces. In our case, the leaves have Riemannian metrics, which vary continuously in the direction transverse to the leaves. (Surface) laminations occur in other areas of mathematics, for example as inverse limits of sequences of finite covers of a fixed compact surface, or as objects obtained by inductively splitting open sheets in a branched surface (the latter can easily occur as attractors of certain kinds of partially hyperbolic dynamical systems). One well-known example is sometimes called the (punctured) solenoid; its Teichmüller theory is studied by Penner and Šarić  (question: does anyone know how to do a “\acute c” in wordpress? update 11/6: thanks Ian for the unicode hint).

A lamination is said to be minimal if every leaf is dense. In our context this means that for every compact region in and every there is a so that every ball in of radius contains a subset which is -close to in the Gromov-Hausdorff metric. In other words, every “local feature” of that appears somewhere, appears with definite density to within any desired degree of accuracy. Consequently, such features will “almost” appear, with the same definite density, in every other leaf of , and therefore is in the closure of each . Since is (in) the closure of , this implies that every leaf is dense, as claimed.

In a Riemann surface lamination, the conformal type of every leaf is well-defined. If some leaf is elliptic, then necessarily that leaf is a sphere. So if the lamination is minimal, it is equal to a single closed surface. If every leaf is hyperbolic, then each leaf admits a unique hyperbolic metric in its conformal class (i.e. each leaf can be uniformized), and Candel showed that this family of hyperbolic metrics varies continuously in . Étienne Ghys asked whether there is an example of a minimal Riemann surface lamination in which some leaves are conformally parabolic, and others are conformally hyperbolic. It turns out that the answer to this question is yes; Richard Kenyon found an example, which I will now describe.

The lamination in question has exactly one hyperbolic leaf, which is topologically a -times punctured sphere. Every other leaf is an infinite cylinder — i.e. it is conformally the punctured plane . Since the lamination is minimal, to describe the lamination, one just needs to describe one leaf. This leaf will be obtained as the boundary of a thickened neighborhood of an infinite planar graph, which is defined inductively, as follows.

Let be the planar “Greek cross” as in the following figure:

Inductively, if we have defined , define by attaching four copies of to the extremities of . The first few examples are illustrated in the following figure:

The limit is a planar tree with exactly four ends; the boundary of a thickened tubular neighborhood is conformally equivalent to a sphere with four points removed, which is hyperbolic. Every unbounded sequence of points in has a subsequence which escapes out one of the ends. Hence every other leaf in the lamination this defines has exactly two ends, and is conformally equivalent to a punctured plane, which is parabolic.

The header image is a very similar construction in -dimensional space, where the initial seed has six legs along the coordinate axes instead of four; some (quite large) approximation was then rendered in povray.

When I was in graduate school, I was very interested in the (complex) geometry of Riemann surface laminations, and wanted to understand their deformation theory, perhaps with the aim of using structures like taut foliations and essential laminations to hyperbolize -manifolds, as an intermediate step in an approach to the geometrization conjecture (now a theorem of Perelman). I know that at one point Sullivan was quite interested in such objects, as a tool in the study of Julia sets of rational functions. I have the impression that they are not studied so much these days, but I would be happy to be corrected.

Bridgeman’s orthospectrum identity

Martin Bridgeman gave a nice talk at Caltech recently on his discovery of a beautiful identity concerning orthospectra of hyperbolic surfaces (and manifolds of higher dimension) with totally geodesic boundary. The -dimensional case is (in my opinion) the most beautiful, and I would like to take a post to explain the identity, and give a derivation which is slightly different from the one Martin gives in his paper. There are many other things one could say about this identity, and its relation to other identities that turn up in the theory of hyperbolic manifolds (and elsewhere); I hope to get to this in a later post.

Let be a hyperbolic surface with totally geodesic boundary. An orthogeodesic is a geodesic segment properly immersed in , which is perpendicular to at its endpoints. The set of orthogeodesics is countable, and their lengths are proper. Denote these lengths by (with multiplicity). The identity is:

where is the Rogers’ dilogarithm function (to be defined in a minute). Treating this function as a black box for the moment, the identity has the form a term depending only on the topology of . The proof is very, very short and elegant. By the Gauss-Bonnet theorem, the term on the right is equal to of the volume of the unit tangent bundle of . Almost every tangent vector on can be exponentiated to a geodesic on which intersects the boundary in finite forward and backward time (eg. by ergodicity of the geodesic flow on a closed hyperbolic surface obtained by doubling). If is such a tangent vector, and is the associated geodesic arc, then is homotopic keeping endpoints on to a unique orthogeodesic (which is the unique length minimizer in this relative homotopy class). The volume of the set of associated to a given orthogeodesic can be computed as follows. Lift to the universal cover, where it is the crossbar of a letter “H” whose vertical lines are lifts of the geodesics it ends on. Any lifts to a unique geodesic segment in the universal cover with endpoints on the edges of the H. So the volume of the set of such depends only on , giving rise to the explicit formula for . qed.

That’s it — that’s the whole proof! . . . modulo some calculations, which we now discuss.

The “ordinary” polylogarithms are defined by Taylor series

which converges for , and extends by analytic continuation. Taking derivatives, one sees that they satisfy , thereby giving rising to integral formulae. is the familiar geometric series , so and

The Rogers dilogarithm is then given by the formula for real . One sees that the Rogers dilogarithm is obtained by symmetrizing the integrand for the integral expression for under the involution :

Martin derives his identity by direct calculation, but in fact this calculation can be simplified a bit by some hyperbolic geometry. Consider an ideal quadrilateral (whose unit tangent bundle has area ) with one pair of opposite sides that are distance apart. Join opposite vertices in pairs to decompose the quadrilateral into four triangles, each with one non-ideal point:

In the (schematic) picture, suppose the two edges of the H are the left and right side (call them and ) and the other two edges are and . Similarly, call the four triangles depending on which edge of the quadrilateral they bound. The triangle is colored gray in the figure. We secretly identify this figure with the upper half-plane, in such a way that the ideal vertices are (in circular order) , where are the ideal vertices of the gray triangle. Call the (hyperbolic) angle of the gray triangle at its vertex, so . Moreover, it turns out that where is the distance between and . We will compute implicitly as a function of , and show that it is a multiple of the Rogers dilogarithm function, thus verifying Bridgeman’s identity.

Every vector in exponentiates to a (bi-infinite) geodesic , and we want to compute the volume of the set of vectors for which the corresponding geodesic intersects both and . The point of the decomposition is that for in (say), the geodesic intersects whenever it intersects , so we only need to compute the volume of the in for which intersects . Similarly, we only need to compute the volume of the in for which intersects . For in , we compute the volume of the which do not intersect (since these are exactly the ones that intersect both and ), and similarly for .

These volumes can be expressed in terms of integrals of harmonic functions. Let denote the harmonic function on the disk which is on the arc of the circle bounded by , and on the rest of the circle. This function at each point is equal to times the visual angle (i.e. the length in the unit tangent circle) subtended by the given arc of the circle, as seen from the given point in the hyperbolic plane. Define similarly. Then the total volume we need to compute is equal to

(here we have identified by symmetry, and similarly for the other pair of terms). Let us approach this a bit more systematically. If denotes the angle at the nonideal vertex of triangle , we denote , and . The integral we want to evaluate can be expressed easily in terms of explicit rational multiples of , and the function . These functions satisfy obvious identities:

and

where the last identity comes by observing that we are integrating a certain function over an ideal triangle, and observing that the average of this function under the symmetries of the ideal triangle is equal to the constant function . In particular, we see that we can express everything in terms of . After some elementary reorganization, we see that the contribution to the volume of the unit tangent bundle of the surface associated to this particular orthogeodesic is

To compute , it makes sense to move to the upper half-space model, and move the endpoints of the interval to and . The harmonic function is equal to on the negative real axis, and on the positive real axis. It takes the value on the line . The area form in the hyperbolic metric is proportional to the Euclidean area form, with constant . In other words, we want to integrate over the region indicated in the figure, where the nonideal angle is , and the base point is :

If we normalize so that the circular arc is part of the semicircle from to , then the real projection of the vertical lines in the figure are and . There is no elementary way to evaluate this integral, so instead we evaluate its derivative as a function of where as before, . This is the definite integral

Integrating by parts gives . This evaluates to

Thinking of as a function of , we get

Comparing values at we see that and the identity is proved.

Well, OK, this is not terribly simple, but a posteriori it gives a way to express the Rogers dilogarithm as a sum of integrals of very simple harmonic functions over hyperbolic triangles, which is a nice geometric way to think of it.

(Update 10/30): This paper by Dupont and Sah relates Rogers dilogarithm to volumes of -simplices, and discusses some interesting connections to conformal field theory and lattice model calculations. I feel like a bit of a dope, since I read this paper while I was in graduate school more than a dozen years ago, but forgot all about it until I was cleaning out my filing cabinet this morning. They cite an older paper of Dupont for the explicit calculations; these are somewhat tedious and unenlightening; however, he does manage to show that the Rogers dilogarithm is characterized by the Abel identity. In other words,

Lemma A.1 (Dupont): Let be a three times differentiable function satisfying

for all . Then there is a real constant such that where is the Rogers dilogarithm (up to an additive constant).

Nevertheless, they don’t seem to have noticed the formula in terms of integrals of harmonic functions over hyperbolic triangles. Perhaps this is also well-known. Do any readers know?

Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface

Hermann Amandus Schwarz (1843-1921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the same time intimately tied to explicit and practical calculation.

I learned about Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface as three quite separate mathematical objects, and I was very surprised to discover firstly that they had all been discovered by the same person, and secondly that they form parts of a consistent mathematical narrative, which I will try to explain in this post to the best of my ability. There is an instructive lesson in this example (for me), that we tend to mine the past for nuggets, examples, tricks, formulae etc. while forgetting the points of view and organizing principles that made their discovery possible. Another teachable example is that of Dehn’s “invention” of combinatorial (infinite) group theory, as a natural branch of geometry; several generations of followers went about the task of reformulating Dehn’s insights and ideas in the language of algebra, “generalizing” them and stripping them of their context, before geometric and topological methods were reintroduced by Milnor, Schwarz (a different one this time), Stallings, Thurston, Gromov and others to spectacular effect (note: I have the second-hand impression that the geometric point of view in group theory (and every other subject) was never abandoned in the Soviet Union).

Schwarz’s minimal surface (also called “Schwarz’s D surface”, and sometimes “Schwarz’s H surface”) is an extraordinarily beautiful triply-periodic minimal surface of infinite genus that is properly embedded in . According to Nitsche’s excellent book (p.240), this minimal surface closely resembles the separating wall between inorganic and organic materials in the skeleton of a starfish. The basic building block of the surface can be described as follows. If the vertices of a cube are -colored, the black vertices are the vertices of a regular tetrahedron. Let denote the quadrilateral formed by four edges of this tetrahedron; then a fundamental piece of Schwarz’s surface is a minimal disk spanning :

The surface may be “analytically continued” by rotating through an angle around each boundary edge. Six copies of fit smoothly around each vertex, and the resulting surface extends (triply) periodically throughout space.

The symmetries of enable us to give it several descriptions as a Riemann surface. Firstly, we could think of as a polygon in the hyperbolic plane with four edges of equal length, and angles . Twelve copies of can be assembled to make a hyperbolic surface of genus . Thinking of a surface of genus as the boundary of a genus handlebody defines a homomorphism from to , thought of as ; the cover associated to the kernel is (conformally) the triply periodic Schwarz surface, and the deck group acts on as a lattice (of index in the face-centered cubic lattice).

Another description is as follows. Since the deck group acts by translation, the Gauss map from to factors through a map . The map is injective at each point in the interior or on an edge of a copy of , but has an order branch point at each vertex. Thus, the map is a double-branched cover, with one branch point of order at each vertex of a regular inscribed cube. This leads one to think (like a late 19th century mathematician) of as the Riemann surface on which a certain multi-valued function on is single-valued. Under stereographic projection, the vertices of the cube map to the eight points where . These eight points are the roots of the polynomial , so we may think of as the hyperelliptic Riemann surface defined by the equation ; equivalently, as the surface on which the multi-valued (on ) function is single-valued.

The function is known as the Weierstrass function associated to , and an explicit formula for the co-ordinates of the embedding were found by Enneper and Weierstrass. After picking a basepoint (say ) on the sphere, the coordinates are given by integration:

The integral in each case depends on the path, and lifts to a single-valued function precisely on .

Geometrically, the three coordinate functions are harmonic functions on . This corresponds to the fact that minimal surfaces are precisely those with vanishing mean curvature, and the fact that the Laplacian of the coordinate functions (in terms of isothermal parameters on the underlying Riemann surface) can be expressed as a nonzero multiple of the mean curvature vector. A harmonic function on a Riemann surface is the real part of a holomorphic function, unique up to a constant; the holomorphic derivative of the (complexified) coordinate functions are therefore well-defined, and give holomorphic -forms which descend to (since the deck group acts by translations). These -forms satisfy the identity (this identity expresses the fact that the embedding of into via these functions is conformal). The (composition of the) Gauss map (with stereographic projection) can be read off from the , and as a meromorphic function on , it is given by the formula . Define a function on by the formula . Then are the coordinates of a rational map from into which extends to a map into , by sending each zero of to in the at infinity. Symmetry allows us to identify the image with the hyperelliptic embedding from before, and we deduce that . Solving for we obtain the integrands in the formulae above.

In fact, any holomorphic function on a domain in defines a (typically immersed with branch points) minimal surface, by the integral formulae of Enneper-Weierstrass above. Suppose we want to use this fact to produce an explicit description of a minimal surface bounded by some explicit polygonal loop in . Any minimal surface so obtained can be continued across the boundary edges by rotation; if the angles at the vertices are all of the form the resulting surface closes up smoothly around the vertices, and one obtains a compact abstract Riemann surface tiled by copies of the fundamental region, together with a holonomy representation of into . Sometimes the image of this representation in the rotational part of is finite, and one obtains an infinitely periodic minimal surface as in the case of Schwarz’s surface. A fundamental tile in can be uniformized as a hyperbolic polygon; equivalently, as a region in the upper half-plane bounded by arcs of semicircles perpendicular to the real axis. Since the edges of the loop are straight lines, the image of this hyperbolic polygon under the Gauss map is a region in also bounded by arcs of round circles; thus Schwarz’s study of minimal surfaces naturally led him to the problem of how to explicitly describe conformal maps between regions in the plane bounded by circular arcs. This problem is solved by the Schwarz-Christoffel transformation, and its generalizations, with help from the Schwarzian derivative.

Note that if and are two such regions, then a conformal map from to can be factored as the product of a map uniformizing as the upper half-plane, followed by the inverse of a map uniformizing as the upper half-plane. So it suffices to find a conformal map when the domain is the upper half plane, decomposed into intervals and rays that are mapped to the edges of a circular polygon . Near each vertex, can be moved by a fractional linear transformation to (part of) a wedge, consisting of complex numbers with argument between and , where is the angle at . The function uniformizes the upper half-plane as such a wedge; however it is not clear how to combine the contributions from each vertex, because of the complicated interaction with the fractional linear transformation. The fundamental observation is that there are certain natural holomorphic differential operators which are insensitive to the composition of a holomorphic function with groups of fractional linear transformations, and the uniformizing map can be expressed much more simply in terms of such operators.

For example, two functions that differ by addition of a constant have the same derivative: . Functions that differ by multiplication by a constant have the same logarithmic derivative: . Putting these two observations together suggest defining the nonlinearity of a function as the composition . This has the property that for any constants . Under inversion the nonlinearity transforms by . From this, and a simple calculation, one deduces that the operator is invariant under inversion, and since it is also invariant under addition and multiplication by constants, it is invariant under the full group of fractional linear transformations. This combination is called the Schwarzian derivative; explicitly, it is given by the formula . Given the Schwarzian derivative , one may recover the nonlinearity by solving the Ricatti equation . As explained in this post, solutions of the Ricatti equation preserve the projective structure on the line; in this case, it is a complex projective structure on the complex line. Equivalently, different solutions differ by an element of , acting by fractional linear transformations, as we have just deduced. Once we know the nonlinearity, we can solve for by , the usual solution to a first order linear inhomogeneous ODE. The Schwarzian of the function is . The advantage of expressing things in these terms is that the Schwarzian of a uniformizing map for a circular polygon with angles at the vertices has the form of a rational function, with principal parts , where the and the and depend (unfortunately in a very complicated way) on the edges of (for the ugly truth, see Nehari, chapter 5). To see this, observe that the map has an order two pole near finitely many points (the preimages of the vertices of under the uniformizing map) but is otherwise holomorphic. Moreover, it can be analytically continued into the lower half plane across the interval between successive , by reflecting the image across each circular edge. After reflecting twice, the image of is transformed by a fractional linear transformation, so has an analytic continuation which is single valued on the entire Riemann sphere, with finitely many isolated poles, and is therefore a rational function! When the edges of the polygon are straight, a simpler formula involving the nonlinearity specializes to the “familiar” Schwarz-Christoffel formula.

(Update 10/22): In fact, I went to the library to refresh myself on the contents of Nehari, chapter 5. The first thing I noticed — which I had forgotten — was that if is the uniformizing map from the upper half-plane to a polygon with spherical arcs, then is real-valued on the real axis. Since it is a rational function, this implies that its nonsingular part is actually a constant; i.e.

where is as above, and are real constants (which satisfy some further conditions — really see Nehari this time for more details).

The other thing that struck me was the first paragraph of the preface, which touches on some of the issues I alluded to above:

In the preface to the first edition of Courant-Hilbert’s “Methoden der mathematischen Physik”, R. Courant warned against a trend discernible in modern mathematics in which he saw a menace to the future development of mathematical analysis. He was referring to the tendency of many workers in this field to lose sight of the roots of mathematical analysis in physical and geometric intuition and to concentrate their efforts on the refinement and the extreme generalization of existing concepts.

Instead of using a word like “menace”, I would rather take this as a lesson about the value of returning to the points of view that led to the creation of the mathematical objects we study every day; which was (to some approximation) the point I was trying to illustrate in this post.

Quasimorphisms from knot invariants

Last week, Michael Brandenbursky from the Technion gave a talk at Caltech on an interesting connection between knot theory and quasimorphisms. Michael’s paper on this subject may be obtained from the arXiv. Recall that given a group , a quasimorphism is a function for which there is some least real number (called the defect) such that for all pairs of elements there is an inequality . Bounded functions are quasimorphisms, although in an uninteresting way, so one is usually only interested in quasimorphisms up to the equivalence relation that if the difference is bounded. It turns out that each equivalence class of quasimorphism contains a unique representative which has the extra property that for all and . Such quasimorphisms are said to be homogeneous. Any quasimorphism may be homogenized by defining (see e.g. this post for more about quasimorphisms, and their relation to stable commutator length).

Many groups that do not admit many homomorphisms to nevertheless admit rich families of homogeneous quasimorphisms. For example, groups that act weakly properly discontinuously on word-hyperbolic spaces admit infinite dimensional families of homogeneous quasimorphisms; see e.g. Bestvina-Fujiwara. This includes hyperbolic groups, but also mapping class groups and braid groups, which act on the complex of curves.

Michael discussed another source of quasimorphisms on braid groups, those coming from knot theory. Let be a knot invariant. Then one can extend to an invariant of pure braids on strands by where , and the “hat” denotes plat closure. It is an interesting question to ask: under what conditions on is the resulting function on braid groups a quasimorphism?

In the abstract, such a question is probably very hard to answer, so one should narrow the question by concentrating on knot invariants of a certain kind. Since one wants the resulting invariants to have some relation to the algebraic structure of braid groups, it is natural to look for functions which factor through certain algebraic structures on knots; Michael was interested in certain homomorphisms from the knot concordance group to . We briefly describe this group, and a natural class of homomorphisms.

Two oriented knots in the -sphere are said to be concordant if there is a (locally flat) properly embedded annulus in with and . Concordance is an equivalence relation, and the equivalence classes form a group, with connect sum as the group operation, and orientation-reversed mirror image as inverse. The only subtle aspect of this is the existence of inverses, which we briefly explain. Let be an arbitrary knot, and let denote the mirror image of with the opposite orientation. Arrange in space so that they are symmetric with respect to reflection in a dividing plane. There is an immersed annulus in which connects each point on to its mirror image on , and the self-intersections of this annulus are all disjoint embedded arcs, corresponding to the crossings of in the projection to the mirror. This annulus is an example of what is called a ribbon surface. Connect summing to by pushing out a finger of each into an arc in the mirror connects the ribbon annulus to a ribbon disk spanning . A ribbon surface (in particular, a ribbon disk) can be pushed into a (smoothly) embedded surface in a -ball bounding . Puncturing the -ball at some point on this smooth surface, one obtains a concordance from to the unknot, as claimed.

The resulting group is known as the concordance group of knots. Since connect sum is commutative, this group is abelian. Notice as above that a slice knot — i.e. a knot bounding a locally flat disk in the -ball — is concordant to the unknot. Ribbon knots (those bounding ribbon disks) are smoothly slice, and therefore slice, and therefore concordant to the trivial knot. Concordance makes sense for codimension two knots in any dimension. In higher even dimensions, knots are always slice, and in higher odd dimensions, Levine found an algebraic description of the concordance groups in terms of (Witt) equivalence classes of linking pairings on a Seifert surface; (some of) this information is contained in the signature of a knot.

Let be a knot (in for simplicity) with Seifert surface of genus . If are loops in , define to be the linking number of with , which is obtained from by pushing it to the positive side of . The function is a bilinear form on , and after choosing generators, it can be expressed in terms of a matrix (called the Seifert matrix of ). The signature of , denoted , is the signature (in the usual sense) of the symmetric matrix . Changing the orientation of a knot does not affect the signature, whereas taking mirror image multiplies it by . Moreover, if are Seifert surfaces for , one can form a Seifert surface for for which there is some sphere that intersects in a separating arc, so that the pieces on either side of the sphere are isotopic to the , and therefore the Seifert matrix of can be chosen to be block diagonal, with one block for each of the Seifert matrices of the ; it follows that . In fact it turns out that is a homomorphism from to ; equivalently (by the arguments above), it is zero on knots which are topologically slice. To see this, suppose bounds a locally flat disk in the -ball. The union is an embedded bicollared surface in the -ball, which bounds a -dimensional Seifert “surface” whose interior may be taken to be disjoint from . Now, it is a well-known fact that for any oriented -manifold , the inclusion induces a map whose kernel is Lagrangian (with respect to the usual symplectic pairing on of an oriented surface). Geometrically, this means we can find a basis for the homology of (which is equal to the homology of ) for which half of the basis elements bound -chains in . Let be obtained by pushing off in the positive direction. Then chains in and chains in are disjoint (since and are disjoint) and therefore the Seifert matrix of has a block form for which the lower right block is identically zero. It follows that also has a zero lower right block, and therefore its signature is zero.

The Seifert matrix (and therefore the signature), like the Alexander polynomial, is sensitive to the structure of the first homology of the universal abelian cover of ; equivalently, to the structure of the maximal metabelian quotient of . More sophisticated “twisted” and signatures can be obtained by studying further derived subgroups of as modules over group rings of certain solvable groups with torsion-free abelian factors (the so-called poly-torsion-free-abelian groups). This was accomplished by Cochran-Orr-Teichner, who used these methods to construct infinitely many new concordance invariants.

The end result of this discussion is the existence of many, many interesting homomorphisms from the knot concordance group to the reals, and by plat closure, many interesting invariants of braids. The connection with quasimorphisms is the following:

Theorem(Brandenbursky): A homomorphism gives rise to a quasimorphism on braid groups if there is a constant so that , where denotes -ball genus.

The proof is roughly the following: given pure braids one forms the knots , and . It is shown that the connect sum bounds a Seifert surface whose genus may be universally bounded in terms of the number of strands in the braid group. Pushing this Seifert surface into the -ball, the hypothesis of the theorem says that is uniformly bounded on . Properties of then give an estimate for the defect; qed.

It would be interesting to connect these observations up to other “natural” chiral, homogeneous invariants on mapping class groups. For example, associated to a braid or mapping class one can (usually) form a hyperbolic -manifold which fibers over the circle, with fiber and monodromy . The -invariant of is the signature defect where is a -manifold with with a product metric near the boundary, and is the first Pontriagin form on (expressed in terms of the curvature of the metric). Is a quasimorphism on some subgroup of (eg on a subgroup consisting entirely of pseudo-Anosov elements)?