Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In this note I first review the basic theory of the Thurston norm, and then present an explicit visualization of two distinct fibrations of the Whitehead link complement. NB: in what follows all surfaces are assumed to be compact and oriented, and homology always will have real coefficients. 1 Surface bundles over S 1 Consider the following simple construction. Let S be a surface, and ϕ : S S be a homeomorphism. The mapping torus of ϕ is the 3-manifold M ϕ obtained by gluing the ends of S [0, 1] together by ϕ. The canonical projection M ϕ S 1 gives M ϕ the structure of a fiber bundle over the circle with fiber S, and the map ϕ is called the monodromy of the bundle. One might ask: are there other maps M ϕ S 1 which realize M ϕ as a surface bundle over the circle? The answer often turns out to be yes and in fact that there are infinitely many different such maps! Here different means that the fibers of these maps represent different relative homology classes in H 2 (M ϕ, M φ ; R). To illustrate this surprising occurrence, here is an example which I found in Benson Farb s notes-in-progress [Fa]. Example 1.1. Let S be the surface of genus 2, and id : S S be the identity map. Then M id = S S 1 is trivially a surface bundle over the circle. We will construct a new fibration by performing surgery on the fibers of this trivial bundle. Figure 1 The bundle S S 1 can be visualized as a thickened version of S, where the 2 boundary components are glued together by id. Take 2 fibers and cut each along the curve shown in Figure 1. Now glue the left boundary component of each fiber to the right boundary component of the other. Cross-sections before and after this surgery are shown in Figure 2. 1
(a) (b) Figure 2: The thin lines in (a) and (b) denote the fibers before and after surgery, respectively. After surgery, the resulting surface of genus 3 is the fiber of a new bundle, whose circle direction is the same and whose monodromy is as follows. If one draws the surface of genus 3 such that its genera lie on a line, the monodromy is rotation by 180 degrees about an axis which travels through the center genus. This example is easily generalizable, and in particular we see that S S 1 has the structure of a surface bundle over the circle with fibers of arbitrarily high genus. 2 The Thurston norm Let M be a compact, oriented 3-manifold, with or without boundary. The possibility of many fiberings of M over S 1, à la Example 1.1, can be well understood via the theory of the Thurston norm, which is a pseudonorm x defined on the finite-dimensional real vector space H 2 (M, M). The pseudonorm is simple to define but, as we will see below, has remarkable properties. For more details and proofs of the following unjustified claims, the reader should consult any of Candel and Conlon [CC], Calegari [Ca], or Thurston s original paper [Th1]. Let S M be an embedded surface. Following Thurston, we define χ (S) = max{0, χ(s)} if S is connected, and χ (S) = i χ (S i ) otherwise, where the sum is over the connected components of S. Now let α H 2 (M, M) be an integral class. We can always find an embedded surface which represents the class α, and we define x(α) = min [S]=α χ (S) 2
where the minimum is taken over all embedded surfaces S with [S] = α. It is not hard to show that x satisfies the triangle inequality. We can extend x to every rational class in H 2 (M, M) by linearity on lines through 0, and since the result is Lipschitz, x can be continuously extended to all of H 2 (M, M). The extension is convex and linear on lines through 0 by continuity. The first thing to note is that if (M, M) possesses an essential sphere or incompressible surface of nonnegative Euler characteristic, then x is strictly a pseudonorm. However, this is the only obstruction to x s normhood: the linear subspace on which x vanishes is in fact spanned by the classes represented by surfaces of nonnegative Euler characteristic. Now suppose that (M, M) is irreducible and atoroidal, so that x is a norm. By definition x is integer-valued on the integer lattice in H 2, and this fact can be used to show that the unit ball B of x is a finite-sided polyhedron whose vertices lie at rational points. This polyhedron B encodes much information about the various ways that M fibers over the circle: Theorem 2.1 (Thurston). There is a collection of faces of B, called fibered faces, such that α H 2 is represented by a fiber if and only if the ray tα intersects B in the interior of a fibered face. In particular, homology classes corresponding to vertices of B cannot be represented by fibers. The collection of fibered faces is possibly empty, but the preceding discussion implies that if one fibering S M S 1 exists, then M fibers in infinitely many distinct ways. This is an interesting phenomenon, explicit examples of which seem to be scarce. Example 1.1 certainly demonstrates multiple fibering, but from the perspective of the Thurston norm it is unsatisfactory, since the 3-manifold in that example contained many incompressible tori. Indeed, the surgery we performed was actually an oriented sum of 2 fibers with an incompressible torus. For the remainder of this note we consider only one 3-manifold, for which the unit ball of x is a finite-sided polyhedron, with the aim of visualizing 2 distinct ways in which it fibers. The word visualization is perhaps vague, but here I use it to mean being able to imagine one fiber sliding along in the circle direction of a fibration, cutting out the entire manifold and eventually returning to itself. Said another way, a satisfactory visualization would identify a cross-section of the suspension flow of some fibration, and enable one to mentally see the cross-section being carried along and back again by the suspension flow. 3 The Whitehead link 3.1 Computing the norm Figure 3: The Whitehead link L = L 1 L 2. 3
Let N be a small open tubular neighborhood of the Whitehead link L = L 1 L 2 S 3, oriented as shown in Figure 3. By deleting N from S 3, we obtain a compact, oriented 3-manifold with boundary, M := S 3 \ N. We would like to compute the unit ball of x on H 2 (M, M), which is isomorphic to H 2 (S 3, N) by excision. Note that we have H 2 (S 3, N) = H 1 (L) = Z n because the exact sequence of the pair (S 3, N) contains 0 = H 2 (S 3 ) H 2 (S 3, N) H 1 (N) H 1 (S 3 ) = 0 and H 1 (N) = H 1 (L) = Z Z. By unwrapping the definition of the boundary map, we see that this isomorphism associates to a relative 2-cycle α of (S 3, N) the vector in Z n whose i-th entry is the signed number of times the boundary of α winds around N. Therefore it makes sense to write x(a 1 L 1 + + a n L n ), where a 1,... a n R, and we will understand that this means x 1 (a 1 L 1 + + a n L n ). Neither component bounds a disk which is not punctured by the other, because L is not the trivial link of 2 components. Indeed, L is not tricolorable. Since the linking number of L is 0, any surface bounded by either component of L is punctured an even number of times by the other. In particular, neither component bounds once-punctured disk, so x(l i ) 1 for i = 1, 2. Since each component bounds a twice-punctured disk (this is made more or less obvious by Figure 3), x(l i ) 1 and thus x(l i ) = 1 for i = 1, 2. Observe that by rotating L 180 degrees around a vertical axis, we maintain the orientation of L 1 while reversing that of L 2. Hence any spanning surface for L gives rise to one of the same Euler characteristic for the link obtained from L by reversing the orientation of L 2. Therefore x(l 1 + L 2 ) = x(l 1 L 2 ). (3.1) Figure 4: A genus 1 Seifert surface Σ for the Whitehead link. Using Seifert s algorithm on the projection of L in Figure 3, we obtain the Seifert surface Σ shown in Figure 4. Computing Euler characteristic shows that this surface is homeomorphic to a torus minus two discs. Therefore x(l 1 + L 2 ) 2. Since the Euler characteristic of a surface is congruent to its number of boundary components modulo 2, x(l 1 +L 2 ) = 2 or 0. If x(l 1 +L 2 ) = 0, then 1 = x(l 1 ) = x( 1 2 (L 1 + L 2 ) + 1 2 (L 1 L 2 )) x(l 1 + L 2 ) 2 = 0, + x(l 1 L 2 ) 2 4
a contradiction. Therefore x(l 1 +L 2 ) = x(l 1 +L 2 ) = 2, and it follows by linearity on rays through the origin that the boundary of the unit ball contains the points (±1, 0), (0, ±1), (± 1 2, ± 1 2 ). The only convex polygon with these 8 points on its boundary is the square with vertices (0, ±1) and (±1, 0), so we have successfully computed the unit ball of x. Remark 3.2. Note that this shows that the surface shown in Figure 4 has minimal genus, so in particular the genus of the Whitehead link is 1. We claim that Σ is a fiber over S 1. However, this is not obvious from our current picture, so we turn to a different picture of Thurston s. 3.2 A decomposition of M In Chapter 3 of his notes [Th2], Thurston observes that S 3 \ L can be realized as the quotient of an octahedron with its vertices deleted where the faces are identified as shown in Figure 5(a). The images of the octahedron s faces form the 2-complex shown in Figure 5(b) (if 5(b) does not make much sense to you, it may help to remember that the link is not there). (a) (b) Figure 5: In (a) the vertices are deleted. The 2-complex in (b) shows the images of the faces in S 3 \ L under this identification. Thurston s observation allowed him to give S 3 \L a hyperbolic structure, but it allows us to see Σ sitting inside an octahedron once we know how it intersects the faces. This data can be obtained by mentally superimposing the pictures in Figures 4 and 5(b), a nice exercise in 3-dimensional visualization. We see that Σ can be isotoped so that the intersections are as shown in Figure 6. The arcs a, b, c, d, and I divide Σ into a bigon, triangle, and pentagon with all vertices deleted. We can draw a, b, c, and d on the surface of our octahedron as shown in Figure 7(a). There is a unique way, up to homotopy rel boundary, to draw in a bigon, triangle, and pentagon with the appropriate boundaries in the interior of the octahedron, which is shown in Figure 7(b). With a little thought, one sees from Figure 7 how the pieces of Σ can flow in M along a vector field transverse to Σ until the picture looks the same again, which gives us the desired fibration. Moreover, we can simultaneously visualize a distinct fibration by observing that there is a symmetry of the manifold corresponding to rotation of the octahedron about the vertical axis by 180 degrees. In the language of Section 3.1, the fibers of this new fibration correspond to L 2 L 1. Note that this shows that all the faces of the unit ball of H 2 (S 3, N) are fibered! Some last thoughts: although the example above has the nice properties that the underlying manifold is hyperbolic and its Thurston norm is nondegenerate, it is still lacking in my opinion. 5
Figure 6: The Seifert surface Σ can be arranged in S 3 \ L to intersect the 2-complex in the arcs a, b, c, d, and I. Figure 7: The Seifert surface Σ. The colors have no significance other than helping to show perspective: the blue bigon is behind the red pentagon, which in turn lies behind the yellow triangle. Specifically, it does not have the advantage of easily generalizing to fibers of arbitrarily high genus like Example 1.1. It would be neat to be able to explicitly visualize an infinite family of fibrations of a hyperbolic manifold over S 1. References [Ca] Calegari, Danny. Foliations and the geometry of 3-manifolds. Oxford University Press, 2007. [CC] Candel, Alberto and Conlon, Lawrence. Foliations II. American Mathematical Society, 2003. [Fa] Farb, Benson. Notes on surface bundles over surfaces. Incomplete notes, 2013. [Ha] Hatcher, Allen. Algebraic topology. Cambridge University Press, 2001. 6
[Th1] [Th2] Thurston, William P. A norm for the homology of 3-manifolds. Memoirs of the American Mathematical Society, Volume 59, 1986. Thurston, William P. The geometry and topology of 3-manifolds. Notes, 1980 (typeset in 2002). Available through MSRI at http://library.msri.org/books/gt3m/. Department of Mathematics, Yale University 10 Hillhouse Ave New Haven, CT 06511 michael.landry@yale.edu 7