Thurston’s Eight Geometries

Every closed surface carries a metric of constant curvature: spherical, flat or hyperbolic, according to its Euler characteristic. This is the uniformization theorem, which the two-dimensional Ricci flow proves (5A.5 Uniformization and the Two-Dimensional Ricci Flow). In three dimensions no such statement can hold for a whole manifold; $S^2\times S^1$, for example, has no metric of constant curvature. Thurston's insight was that it holds piece by piece. After the canonical cuts along spheres and tori (10A.3 The Prime Decomposition, 10A.4 Seifert Spaces and the JSJ Decomposition), each piece carries one of exactly eight model geometries. That is the geometrization conjecture, proved by Perelman with the Ricci flow. The Poincaré conjecture is one consequence.

This chapter lists the eight geometries, says which closed manifolds carry each, gives their curvatures, states the conjecture, and proves that it implies the Poincaré conjecture.

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

• define a model geometry and list the eight three-dimensional ones with their isometry groups;
• give a closed manifold carrying each, and compute or recall its curvature;
• explain the geometry of Sol and Nil through their metrics;
• state the geometrization conjecture and the elliptization conjecture;
• prove that geometrization implies the Poincaré conjecture.

5.1Crystals and flat manifolds

5.2Model geometries

A model geometry is a simply connected manifold $X$ with a Lie group $G$ of diffeomorphisms acting transitively, with compact point stabilisers, such that $G$ is maximal among such groups and some discrete subgroup of $G$ has a compact quotient. Then $X$ has a $G$-invariant Riemannian metric, and $G$ acts by its isometries. A manifold carries the geometry $X$ if it is $X/\Gamma$ for a discrete subgroup $\Gamma \subset G$ acting freely.

geometry	$\dim G$	a closed manifold carrying it	curvature
$S^3$	$6$	lens spaces, the Poincaré sphere	constant $+1$
$\mathbb{E}^3$	$6$	the 3-torus, Hantzsche–Wendt	flat
$\mathbb{H}^3$	$6$	the Seifert–Weber space, the Weeks manifold	constant $-1$
$S^2\times\mathbb{R}$	$4$	$S^2\times S^1$, $\mathbb{RP}^3\#\mathbb{RP}^3$	$\operatorname{Ric}$: $(1, 1, 0)$
$\mathbb{H}^2\times\mathbb{R}$	$4$	$\Sigma_g\times S^1$, $g \geq 2$	$\operatorname{Ric}$: $(-1, -1, 0)$
$\widetilde{SL}(2, \mathbb{R})$	$4$	unit tangent bundles of hyperbolic surfaces	mixed signs
Nil	$4$	nontrivial circle bundles over $T^2$	$\operatorname{Ric}$: $(\frac12, -\frac12, -\frac12)$
Sol	$3$	torus bundles over $S^1$ with hyperbolic monodromy	$\operatorname{Ric}$: $(0, 0, -2)$

The curvatures are those of 9A.5 Computing Curvature, for the standard left-invariant metrics (Milnor's frames for Nil and Sol). The three isotropic geometries, with $6$-dimensional isometry groups, have constant curvature. The other five are anisotropic: they have a preferred direction or splitting. Six of the eight (all but $\mathbb{H}^3$ and Sol) are exactly the geometries of Seifert fibred spaces, selected by the sign of the orbifold Euler characteristic and the Euler number, as in the last exercise of 10A.4 Seifert Spaces and the JSJ Decomposition.

The eight are pairwise different. The dimension of the isometry group, constant curvature, and the signature of the Ricci tensor separate them (Exercise 5.4).

5.3Nil and Sol

Nil is the Heisenberg group of upper triangular $3\times3$ matrices with ones on the diagonal. Its left-invariant metric can be written

The "horizontal" directions, where $dz = \frac12(x\,dy - y\,dx)$, do not form the tangent planes of any surface. A horizontal path that projects to a closed loop in the $(x, y)$-plane rises, by the signed area the loop encloses (Exercise 5.5, Figure 5.1). This "corkscrew" is the geometry of a circle bundle with nonzero Euler number: the circles are the $z$-direction, and going around a loop in the base twists you along them.

Sol is $\mathbb{R}^3$ with the metric

Moving up in $z$ stretches the $x$-direction and squeezes the $y$-direction exponentially (Figure 5.2). The translations in $x, y$ and the maps $(x, y, z) \mapsto (e^{-t}x, e^ty, z + t)$ are isometries. Closed Sol manifolds are torus bundles over the circle whose gluing map is a hyperbolic matrix in $SL(2, \mathbb{Z})$, such as $\begin{pmatrix}2 & 1\\ 1 & 1\end{pmatrix}$, which stretches one eigendirection and squeezes the other, exactly the motion along $z$.

5.4The geometrization conjecture

Thurston stated the conjecture in 1982, having proved it for Haken manifolds. The special case for manifolds with finite fundamental group is the elliptization conjecture: a closed 3-manifold with finite $\pi_1$ is a spherical space form $S^3/\Gamma$. Hamilton proposed attacking geometrization with the Ricci flow (10A.7 Geometrization and Ricci Flow), and Perelman's papers of 2002–03 completed the program (12C.4 Geometrization).

5.5History

Fedorov and Schoenflies classified the space groups in 1891; Bieberbach proved his theorems in 1911–12; Hantzsche and Wendt listed the flat 3-manifolds in 1935. Thurston formulated the geometrization conjecture and the eight geometries in the late 1970s, published in his 1982 Bulletin paper, and Scott's 1983 survey gave the classification its standard form. Jeffrey Weeks found the Weeks manifold in his 1985 thesis; Gabai, Meyerhoff and Milley proved in 2009 that it has the smallest volume of any closed orientable hyperbolic 3-manifold.

5.6Exercises

Solution

Dimension $6$: $S^3$, $\mathbb{E}^3$, $\mathbb{H}^3$, separated by the sign of their constant curvature. Dimension $3$: Sol alone. Dimension $4$: $S^2\times\mathbb{R}$ has Ricci signs $(+, +, 0)$ and $\mathbb{H}^2\times\mathbb{R}$ has $(-, -, 0)$. For $\widetilde{SL}(2, \mathbb{R})$, $\mu = (-\frac c2, -\frac c2, a + \frac c2)$ and $\operatorname{Ric} = \big(-c(a + \frac c2), -c(a + \frac c2), \frac{c^2}{2}\big)$: one positive and two equal negative eigenvalues, the same signs as Nil's $(\frac12, -\frac12, -\frac12)$. But the ratio of the positive eigenvalue to the negative ones is $-\frac{c}{2a + c} \in (-1, 0)$, never $-1$ as for Nil, and this ratio is unchanged by scaling. So no metric of one geometry is a rescaling of a metric of the other.

Solution

$\dot z = \frac12(\cos^2t + \sin^2t) = \frac12$, so $z(2\pi) = \pi$. In general $\Delta z = \oint\frac12(x\,dy - y\,dx)$, which by Green's theorem is the signed area enclosed.

Solution

Translations in $x$ and $y$ preserve $g$, because its coefficients depend only on $z$. For the second, $x' = e^{-t}x$, $y' = e^ty$, $z' = z + t$: $e^{2z'}dx'^2 = e^{2z + 2t}e^{-2t}dx^2 = e^{2z}dx^2$, and similarly for $y$. Diagonalise $A$ over $\mathbb{R}$; in its eigenbasis $A$ acts as $\operatorname{diag}(\lambda^{-1}, \lambda)$, which is the second isometry with $t = \log\lambda$ restricted to a horizontal plane. The group generated by the lattice translations (in the eigenbasis coordinates) and this map acts freely and cocompactly; the quotient is the mapping torus of $A$ on $T^2$, a torus bundle over the circle.

Solution

The map is an isometry of $\mathbb{R}^3$ normalising $\mathbb{Z}^3$, and it has no fixed points on $T^3$ because of the half-translation in $x$. Its linear part $\operatorname{diag}(1, -1, -1)$ has determinant $1$, so it preserves orientation and the quotient is orientable. $H_1$ is the abelianisation of the group generated by $\mathbb{Z}^3$ and the involution $\tau$, with $\tau^2$ = translation by $(1, 0, 0)$ and $\tau$ inverting the $y$ and $z$ translations: the relations make $2e_y = 2e_z = 0$ in $H_1$, giving $H_1 = \mathbb{Z}\oplus\mathbb{Z}/2\oplus\mathbb{Z}/2$.

Solution

If $\pi_1(M)$ is finite, $M$ is prime: a nontrivial connected sum has $\pi_1$ a free product, which is infinite unless one factor is trivial, and then (with geometrization applied to that factor, as in the proof) that summand is $S^3$. $M$ is not $S^2\times S^1$ (infinite $\pi_1$), so it is irreducible. There are no incompressible tori, since $\mathbb{Z}^2$ would inject into a finite group. So $M = X/\Gamma$ is geometric with $\Gamma$ finite. Then the universal cover $X$ is compact, so $X = S^3$, and $M = S^3/\Gamma$.