Seifert Spaces and the JSJ Decomposition

The prime decomposition (10A.3 The Prime Decomposition) cuts along spheres. Prime pieces can still be complicated, and the next canonical cut is along tori. The tori that matter are incompressible: they cannot be simplified by cutting along a disc. The JSJ decomposition (Jaco–Shalen, Johannson) says that an irreducible 3-manifold has a canonical, minimal family of incompressible tori that cuts it into pieces of two kinds. One kind is Seifert fibred, filled by circles in a controlled way. The other is atoroidal: no incompressible tori remain except those parallel to the boundary. Thurston's geometrization conjecture predicts that the atoroidal pieces are hyperbolic. The Seifert pieces already carry geometries, read off from the circles and the surface they fibre over (10A.5 Thurston’s Eight Geometries).

This chapter is topology, with no physical examples. Its payoff in the Ricci flow is the thin part. After a long time, the regions of a 3-manifold where the rescaled flow collapses with bounded curvature are graph manifolds, glued from Seifert pieces along tori. Hyperbolic pieces are what remains thick (10A.7 Geometrization and Ricci Flow, 12C.4 Geometrization).

By the end of this chapter you will be able to:

• define incompressible surfaces, and show the boundary torus of a nontrivial knot complement is incompressible;
• describe Seifert fibred spaces through fibred solid tori, exceptional fibres and base orbifolds;
• give Seifert fibrations of lens spaces, the Poincaré sphere, torus knot complements and unit tangent bundles;
• compute the orbifold Euler characteristic of a base and use it, with the Euler number, to predict the geometry;
• state the JSJ decomposition and describe graph manifolds.

4.1Incompressible surfaces

A compact surface $S \neq S^2, D^2$ properly embedded (or embedded) in a 3-manifold $M$ is incompressible if there is no compressing disc: no embedded disc $D \subset M$ with $D\cap S = \partial D$ a curve that does not bound a disc in $S$. For two-sided surfaces this is equivalent, by Dehn's lemma and the loop theorem (10A.3 The Prime Decomposition), to $\pi_1(S) \to \pi_1(M)$ being injective. A torus that compresses can be cut along the disc into a sphere, so in an irreducible manifold compressible tori are inessential: they bound solid tori or lie in balls.

Examples: the torus $T^2\times\{\ast\}$ in $T^3$ is incompressible; the boundary torus of a solid torus is compressible (the meridian disc); the boundary torus of the exterior of a nontrivial knot in $S^3$ is incompressible (Exercise 4.3).

4.2Seifert fibred spaces

Fibred solid tori. For coprime integers $p \geq 1$ and $q$, take the cylinder $D^2\times[0, 1]$ and glue the top to the bottom with a rotation by $2\pi\frac qp$. The vertical segments glue up into circles. The central segment closes into a single circle, the core. Every other point lies on a circle that goes $p$ times around the core before closing (Figure 4.1). This is the fibred solid torus of type $(p, q)$. If $p = 1$, all circles are alike: it is a product $D^2\times S^1$. If $p > 1$, the core is an exceptional fibre of multiplicity $p$.

Collapsing each fibre to a point gives the base $B$, a surface. The images of exceptional fibres are cone points of $B$, with cone angle $\frac{2\pi}{p}$ for multiplicity $p$. In the language of orbifolds, $B$ is a 2-dimensional orbifold, and the fibration is a kind of circle bundle over it. The base orbifold has an orbifold Euler characteristic

and there is a rational Euler number $e$ measuring how twisted the circles are, the analogue of the Euler number of a circle bundle. Peter Scott showed (1983) that these two numbers alone decide which geometry a closed Seifert fibred space carries (Exercise 4.7, 10A.5 Thurston’s Eight Geometries).

Examples.

• The Hopf fibration (10A.1 A Zoo of Three-Manifolds) is a Seifert fibration of $S^3$ over $S^2$ with no exceptional fibres. Since the lens space actions $(z, w) \mapsto (\zeta z, \zeta^qw)$ permute Hopf-type circles, lens spaces are Seifert fibred as well.
• The Poincaré homology sphere is Seifert fibred over $S^2$ with three exceptional fibres of multiplicities $2$, $3$, $5$: its base is the orbifold $S^2(2, 3, 5)$, the quotient of $S^2$ by the rotation group of the icosahedron.
• Torus knot complements. The complement of the $(p, q)$ torus knot is Seifert fibred over a disc with two cone points of orders $p$ and $q$. For the trefoil, $2$ and $3$.
• Unit tangent bundles. For a closed surface $\Sigma$, the unit tangent bundle $UT\Sigma$ is a circle bundle, Seifert fibred with no exceptional fibres, Euler number $\chi(\Sigma)$ (Exercise 4.5). $UTS^2 = \mathbb{RP}^3$; $UT$ of a hyperbolic surface carries the geometry $\widetilde{SL}(2, \mathbb{R})$.
• Products and nilmanifolds. $S^2\times S^1$, $T^3$ and $\Sigma\times S^1$ are trivially fibred. Circle bundles over $T^2$ with nonzero Euler number are the nilmanifolds of 9B.3 Collapsing and Noncollapsing.

Not every manifold is Seifert fibred: the figure-eight knot complement and the Seifert–Weber space are not, being hyperbolic (10A.6 Hyperbolic Three-Manifolds). A Seifert fibred space with infinite fundamental group has a normal infinite cyclic subgroup, generated by a regular fibre, which hyperbolic groups never have.

4.3The JSJ decomposition

William Jaco and Peter Shalen, and independently Klaus Johannson, proved this in 1979 (hence JSJ). It is canonical, like the prime decomposition, and the pieces are of two contrasting kinds (Figure 4.2). A manifold built entirely from Seifert pieces glued along tori is a graph manifold; these include all Seifert fibred spaces and many manifolds that are not. Thurston's hyperbolization theorem for Haken manifolds (manifolds containing an incompressible surface), announced in the late 1970s, showed that atoroidal Haken pieces with infinite fundamental group are hyperbolic. Perelman's work extended this to all atoroidal pieces (10A.7 Geometrization and Ricci Flow).

4.4History

Herbert Seifert introduced his fibred spaces in 1933. Wolfgang Haken's theory of incompressible surfaces dates from the early 1960s, and Friedhelm Waldhausen classified graph manifolds and proved rigidity results for Haken manifolds in the late 1960s. Jaco–Shalen and Johannson published the torus decomposition in 1979. Thurston announced his hyperbolization theorem for Haken manifolds in the late 1970s, and Scott's survey of the eight geometries and Seifert fibrations appeared in 1983.

4.5Exercises

Solution

(a) $c = a\mu + b\lambda$ bounds $D$, so it is $0$ in $H_1(E) = \mathbb{Z}\mu$, with $\lambda$ also $0$ there; hence $a = 0$, and $c$ being a simple essential curve forces $c = \pm\lambda$. (b) $\lambda$ cobounds an annulus with $K$ inside $N(K)$; together with $D$ this gives a disc bounded by $K$, so $K$ is trivial.

Solution

$S^2(2, 3, 5)$: $2 - \frac12 - \frac23 - \frac45 = \frac{1}{30} = \frac{2}{60}$. $S^2(2, 3, 6)$: $2 - \frac12 - \frac23 - \frac56 = 0$. $S^2(2, 3, 7)$: $2 - \frac12 - \frac23 - \frac67 = -\frac{1}{42}$. Disc with $2, 3$: $1 - \frac12 - \frac23 = -\frac16$.

Solution

The map $(x, v) \mapsto$ the matrix with columns $x, v, x\times v$ is a diffeomorphism $UTS^2 \to SO(3)$, and $SO(3) \cong \mathbb{RP}^3$ (8A.5 Lie Groups and Group Actions). The Euler number of the tangent circle bundle counts, with signs, the zeros of a generic section of $T\Sigma$, which by Poincaré–Hopf is $\chi(\Sigma)$. A circle bundle is trivial exactly when its Euler number is zero, so only for $\chi = 0$.

Solution

Following the fibre, each passage through the cylinder rotates the point by $2\pi\frac qp$. Since $\gcd(p, q) = 1$, the rotations by multiples of $2\pi\frac qp$ return to the start first after $p$ steps, so the fibre crosses the disc in $p$ points and winds $p$ times around the core, while the core closes after one passage. For $p = 1$ the rotation is by a multiple of $2\pi$, every fibre closes after one passage, and all fibres are alike.

Solution

$S^2\times S^1$: base $S^2$ ($\chi = 2$), $e = 0$: $S^2\times\mathbb{R}$. $T^3$: base $T^2$, $e = 0$: $\mathbb{E}^3$ (flat). Poincaré sphere: base $S^2(2, 3, 5)$, $\chi^{\mathrm{orb}} = \frac1{30} > 0$, $e \neq 0$ (it is not a product, its $\pi_1$ is finite): $S^3$, consistent with $S^3/I^*$. $UT\Sigma_2$: base $\Sigma_2$, $\chi = -2$, $e = \pm2 \neq 0$: $\widetilde{SL}(2, \mathbb{R})$. Nontrivial circle bundle over $T^2$: Nil. $\Sigma_2\times S^1$: $\mathbb{H}^2\times\mathbb{R}$.