Building Three-Manifolds

The zoo of 10A.1 A Zoo of Three-Manifolds raises a question: is there a systematic way to produce every closed 3-manifold? There are two, and both are constructions from simple pieces. A Heegaard splitting cuts a closed orientable 3-manifold into two handlebodies, solid "doughnuts with $g$ holes", glued along their boundary surfaces, so the whole manifold is encoded by one gluing map of a surface. Dehn surgery removes a solid torus around a knot and glues it back differently. Every closed orientable 3-manifold arises from $S^3$ this way, by surgery on a link.

This chapter is a construction kit, and it has no real-world examples: the figures do the work. The kit matters later for two reasons. Genus-one splittings explain why the lens spaces are the manifolds they are, and surgery descriptions are how topologists name, compare and compute with 3-manifolds. Keep one distinction clear: Dehn surgery here is a topological operation. The surgery of the Ricci flow (12B.4 Surgery) cuts along 2-spheres, not tori, and is a geometric procedure, related to the prime decomposition of 10A.3 The Prime Decomposition.

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

• define handlebodies and Heegaard splittings, and produce a splitting from a triangulation or a Morse function;
• classify the manifolds with Heegaard splittings of genus at most one;
• define Dehn surgery on a knot and compute the first homology of the result;
• state the Lickorish–Wallace theorem and recognise surgery descriptions of $S^3$, $S^2\times S^1$, lens spaces and the Poincaré homology sphere.

2.1Handlebodies and Heegaard splittings

A handlebody of genus $g$, $H_g$, is a closed regular neighbourhood of a connected graph with first Betti number $g$ in $\mathbb{R}^3$: a solid ball with $g$ solid handles attached. Its boundary is the closed orientable surface $\Sigma_g$ of genus $g$. It contains $g$ disjoint meridian discs, properly embedded discs that cut it into a ball (Figure 2.1).

Since any two handlebodies of the same genus are homeomorphic, $M$ is determined by the gluing homeomorphism $\partial H \to \partial H'$, and in fact by where it sends the boundaries of the meridian discs of $H'$: $g$ disjoint curves on $\Sigma_g$. A surface with two such families of curves is a Heegaard diagram.

A Morse function gives the same result (7A.10 Morse Theory). Choose one with a single minimum and a single maximum, all index-1 critical values below all index-2 ones. The sublevel set just above the index-1 values is a ball with 1-handles, a handlebody, and its complement is built from the maximum downwards by 2-handles turned upside down, another handlebody.

Small genus.

• Genus 0. Two balls glued along a sphere: $S^3$, whatever the gluing, because every homeomorphism of $S^2$ extends over the ball (the Alexander trick).
• Genus 1. Two solid tori $V, V'$ glued along their boundary torus. The result depends only on the curve $\gamma \subset \partial V$ to which the meridian of $V'$ is glued, up to isotopy, and $\pi_1 = \mathbb{Z}/p$ where $p$ is the number of times $\gamma$ crosses a meridian disc of $V$ (algebraically) (Exercise 2.4, Figure 2.2). If $p = 1$ the result is $S^3$; if $p = 0$ (meridian to meridian) it is $S^2\times S^1$; otherwise it is a lens space $L(p, q)$. In 10A.1 A Zoo of Three-Manifolds, $S^3$ was split into the solid tori $|z| \leq |w|$ and $|z| \geq |w|$, which is the case $p = 1$.

So the manifolds of Heegaard genus at most one are exactly $S^3$, $S^2\times S^1$ and the lens spaces. The Poincaré homology sphere has genus $2$.

Heegaard splittings are not unique: adding a trivial handle (stabilisation) increases the genus by one. Reidemeister and Singer proved in 1933 that any two splittings of the same manifold become isotopic after enough stabilisations. Friedhelm Waldhausen proved in 1968 that every splitting of $S^3$ is a stabilisation of the genus-0 one.

2.2Dehn surgery

Let $K \subset M$ be a knot, $N(K)$ a closed tubular neighbourhood, a solid torus, and $E = M\setminus\operatorname{int}N(K)$ its exterior. On the boundary torus there are two distinguished curves: the meridian $\mu$, which bounds a disc in $N(K)$, and, for $K \subset S^3$, the longitude $\lambda$, which runs once along $K$ and bounds a surface in $E$. Dehn surgery with coefficient $\frac pq$ glues a solid torus back to $E$ so that its meridian goes to the curve $p\mu + q\lambda$ (coprime $p, q$). The result is $M_K(\frac pq)$. The coefficient $\frac10 = \infty$ gives $M$ back; integral surgeries have $q = 1$.

Surgery on a link does this on each component at once.

W. B. R. Lickorish (1962) proved it by writing the gluing map of a Heegaard splitting as a product of Dehn twists, each realised by a surgery; Andrew Wallace (1960) proved it through 4-dimensional handlebodies. Robion Kirby (1978) found the moves that relate two surgery descriptions of the same manifold, the Kirby calculus.

Examples.

• Surgery on the unknot. Its exterior is a solid torus, so every surgery is a genus-one Heegaard splitting. Coefficient $\frac1n$ gives $S^3$, coefficient $0$ gives $S^2\times S^1$, and coefficient $\frac pq$ gives a lens space with $\pi_1 = \mathbb{Z}/p$ (Exercise 2.6).
• The Poincaré homology sphere is surgery with coefficient $\pm1$ on a trefoil, with the sign matched to the handedness of the trefoil (Figure 2.3). Any $\pm1$ surgery on a knot in $S^3$ gives a homology sphere (Exercise 2.7).
• Knots are determined by their complements (Gordon–Luecke, 1989), and consequently a nontrivial surgery on a nontrivial knot never gives $S^3$. Kronheimer and Mrowka proved in 2004 that no surgery with coefficient $\frac1n$, $n \neq 0$, on a nontrivial knot produces even a simply connected manifold ("Property P"), a statement that no longer needs separate proof once the Poincaré conjecture is known but was proved independently of it.

2.3History

Poul Heegaard introduced his splittings in his 1898 dissertation. Max Dehn defined surgery in 1910, constructing infinitely many homology spheres. Reidemeister and Singer proved the stabilisation theorem in 1933. Wallace (1960) and Lickorish (1962) proved that surgery on links gives every closed orientable 3-manifold, and Kirby's calculus appeared in 1978. Gordon and Luecke's theorem dates from 1989, and Kronheimer and Mrowka's proof of Property P from 2004.

2.4Exercises

Solution

$\pi_1(V') = \mathbb{Z}$ is generated by its longitude, which is the image of a curve on the torus that meets $\gamma$ once; attaching $V'$ amounts to attaching a 2-cell along $\gamma$ and then a 3-cell, so $\pi_1 = \pi_1(V)/\langle\gamma\rangle = \mathbb{Z}/\langle a\rangle$ ($\mu$ is trivial in $\pi_1(V)$). $a = \pm1$ gives a simply connected genus-one manifold, which is $S^3$ (the splitting of 10A.1 A Zoo of Three-Manifolds); $a = 0$ means $\gamma = \pm\mu$ and the result is $S^2\times S^1$, with $\pi_1 = \mathbb{Z}$.

Solution

A handlebody deformation retracts to a graph with Betti number $g$, so $\chi = 1 - g$ and $\pi_1$ is free on $g$ generators (7A.5 Computing π₁). The 1-skeleton is a connected graph with $\chi = V - E$, so its Betti number, and the genus, is $E - V + 1$.

Solution

$S^3$ is two solid tori (10A.1 A Zoo of Three-Manifolds); one is $N(\text{unknot})$, and the other's meridian is the unknot's longitude. Regluing a solid torus along $p\mu + q\lambda$ gives a genus-one splitting. In the exterior $E$, $\pi_1(E) = \mathbb{Z}$ is generated by $\mu$ and $\lambda = 1$, so the result has $\pi_1 = \mathbb{Z}/\langle p\mu + q\lambda\rangle = \mathbb{Z}/p$. For $\frac1n$: $\pi_1 = 1$ and it is $S^3$; for $0$: the meridian goes to $\lambda$, the meridian of the exterior, giving $S^2\times S^1$.

Solution

Gluing in a solid torus adds a 2-cell along $p\mu + q\lambda$ and a 3-cell, so $H_1 = \mathbb{Z}\mu/\langle p\mu + q\cdot0\rangle = \mathbb{Z}/p$. For $p = \pm1$ this is $0$, and $H_2 = 0$, $H_3 = \mathbb{Z}$ by Poincaré duality, so the result is a homology sphere. Surgery $\pm1$ on the unknot gives $S^3$ and on a trefoil gives the Poincaré sphere: same homology, different manifolds, distinguished by $\pi_1$.

Solution

For splittings $M_i = H_i\cup H_i'$, take the connected sum along a ball meeting each splitting surface in a disc; the handlebodies combine by boundary connected sum, $H_1\natural H_2$ and $H_1'\natural H_2'$, of genus $g_1 + g_2$. With $M_2 = S^3$ and its genus-1 splitting, this is a stabilisation of $M_1$'s splitting, of genus $g + 1$. In general it shows $g(M_1\#M_2) \leq g(M_1) + g(M_2)$; Haken's lemma (a reducing sphere can be made to meet a splitting surface in one circle) gives the reverse inequality.