Draft:Geometrization in dimension four

The uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In dimension 3 it is not always possible to assign a geometry to a closed 3-manifold but the resolution of the Geometrization conjecture, proposed by, implies that closed 3-manifolds can be decomposed into geometric ``pieces''. Each of these pieces can have one of 8 possible geometries: spherical $$S^3$$, Euclidean $$\mathbb{E}^3$$, hyperbolic $$\mathbf{H}^3_{\mathbb{R}} $$, Nil geometry $$\mathbf{Nil}^3$$, Sol geometry $$\mathbf{Sol}^3$$, $$\widetilde{\mathrm{SL}_2(\mathbb{R})}$$, and the products $$S^2\times \mathbb{R}$$, and $$\mathbf{H}^2_{\mathbb R}\times\mathbb{R}$$.

In dimension four the situation is more complicated. Not every closed 4-manifold can be uniformized by a Lie group or even decomposed into geometrizable pieces. This follows from unsolvability of the homeomorphism problem for 4-manifolds. But, there is still a classification of 4-dimensional geometries due to Richard Filipkiewicz. These fall into 18 distinct geometries and one infinite family. An in depth discussion of the geometries and the manifolds that afford them is given in Hillman's book. The study of complex structures on geometrizable 4-manifolds was initiated by Wall

The Four Dimensional Geometries
The distinction in to the following classes is somewhat arbitrary, the emphasis has been placed on properties of the fundamental group and the uniformizing Lie group. The classification of the geometries is taken from. . The descriptions of the fundamental groups as well as further information on the 4-manifolds that afford them can be found in Hillman's book

Spherical or compact type
Three geometries lie here, the 4-sphere $$S^4$$, the complex projective plane $$\mathbf{P}^2_\mathbb{C} $$, and a product of two 2-spheres $$S^2\times S^2$$. The fundamental group of any such manifold is finite.

Euclidean type
This is the four dimensional Euclidean space $$\mathbb{E}^4$$. With isometry group $$\mathbb{R}^4\rtimes\mathrm{O}(4)$$. The fundamental group of any such manifold is a Bieberbach group. There are 74 homeomorphism classes of manifolds with geometry $$\mathbb{E}^4$$, 27 orientable manifolds and 47 non-orientable manifolds.

Nilpotent type
There are two geometries of Nilpotent type $$\mathbf{Nil}^4$$ and the reducible geometry $$\mathbf{Nil}^3\times\mathbb{R}$$.

The $$\mathbf{Nil}^4$$ geometry is a 4-dimensional nilpotent Lie group given as the semi-direct product $$\mathbb{R}^3\rtimes_\Theta\mathbb{R}$$, where $$\Theta(t)=\mathrm{diag}[t,t,\frac{1}{2}t^2]$$. The fundamental group of a closed orientable $$\mathbf{Nil}^4$$-manifold is nilpotent of class 3.

For a closed 4-manifold $$M$$ admitting a $$\mathbf{Nil}^3\times\mathbb{R}$$ geometry, there is a finite cover $$M'$$ of $$M$$ such that $$\pi_1 M\cong \Gamma\times\mathbb{Z}$$. Here $$\Gamma$$ is the fundamental group of a 3-dimensional nilmanifold. Thus, every such fundamental group is nilpotent of class 2.

Note that one can always take $$\Gamma$$ above to be one of the following groups $$\Gamma_q := \langle x,y,z\ |\ xz=zx,\ zy=yz,\ xy=z^qyx\rangle$$, where $$q\in\mathbb{Z}$$ is non-zero. These are all fundamental groups of torus bundles over the circle.

Solvable type
There are two unique geometries $$\mathbf{Sol}^4_0$$, and $$\mathbf{Sol}^4_1$$. As well as a countably infinite family $$\mathbf{Sol}^4_{m,n}$$ where $$m,n\geq1$$ are integers.

The $$\mathbf{Sol}^4_0$$-geometry is the Lie group described by the semi-direct product $$\mathbb{R}^3\rtimes_\xi\mathbb{R}$$, where $$\xi(t)=\mathrm{diag}[e^t,e^t,e^{-2t}]$$. The fundamental group of a closed $$\mathbf{Sol}^4_0$$-manifold is a semidirect product $$\mathbb{Z}^3\rtimes_A\mathbb{Z}$$ where $$A\in\mathrm{GL}_3(\mathbb{Z})$$ has one real eigenvalue and two conjugate complex eigenvalues. The fundamental group has Hirsh length equal to 4.

The $$\mathbf{Sol}^4_1$$-geometry is the Lie group described by set of matrices $$\left\{\left[\begin{array}{ccc} 1 & x & z \\ 0 & t & y \\ 0 & 0 & 1\end{array}\right]\ |\ x,y,z,t\in\mathbb{R},\ t>0\right\}$$.

A closed $$\mathbf{Sol}^4_1$$-manifold $$M$$ is a mapping torus of a $$\mathbf{Nil}^3$$-manifold. Its fundamental group is a semidirect product $$\Gamma_q\rtimes\mathbb{Z}$$. The fundamental group has Hirsh length equal to 4.

Define $$f(x)=x^3-mx^2+nx-1$$. If $$m,n$$ are positive integers such that $$0<2\sqrt{n}\leq m < n$$, then $$f(x)$$ has three distinct real roots $$a,b,c$$.

The $$\mathbf{Sol}^4_{m,n}$$-geometry is the Lie group described by the semi-direct product $$\mathbb{R}^3\rtimes_{\Phi_{m,n}}\mathbb{R}$$, where $$\Theta_{m,n}(t)=\mathrm{diag}[e^{at},e^{bt},e^{ct}]$$. The fundamental group of a closed $$\mathbf{Sol}^4_{m,n}$$-manifold is a semidirect product $$\mathbb{Z}^3\rtimes_A\mathbb{Z}$$ where $$A\in\mathrm{GL}_3(\mathbb{Z})$$ has three distinct real eigenvalues. The fundamental group has Hirsh length equal to 4.

Isomorphisms between solvable geometries
Note that when $$n=m$$ that $$\Theta_{n,n}$$ has exactly one eigenvalue. So there is an identification $$\mathbf{Sol}^4_{n,n}=\mathbf{Sol^3}\times\mathbb{R}$$.

We have that $$\mathbf{Sol}^4_{m,n}=\mathbf{Sol}^4_{m',n'}$$ if the roots $$(a,b,c)$$ and $$(a',b',c')$$ satisfy $$\lambda(a,b,c)=(a',b',c')$$ for some real number $$\lambda$$.

A proof of these facts appears in.

Hyperbolic type
There are two geometries here real-hyperbolic 4-space $$\mathbf{H}^4_{\mathbb{R}}$$ and the complex hyperbolic plane $$\mathbf{H}^2_{\mathbb{C}}$$. The fundamental groups of closed manifolds here are word hyperbolic groups.

Product of hyperbolic planes
This is the geometry $$\mathbf{H}^2_{\mathbb{R}}\times \mathbf{H}^2_{\mathbb{R}}$$. Closed manifolds come in two forms here. A $$\mathbf{H}^2_{\mathbb{R}}\times \mathbf{H}^2_{\mathbb{R}}$$-manifold is reducible if it is finitely covered by a direct product of hyperbolic Riemann surfaces. Otherwise it is irreducible. The irreducible manifolds fundamental groups are arithmetic groups by Margulis' arithmeticity theorem.

The tangent space of the hyperbolic plane
This geometry admits no closed manifolds.

Remaining geometries
The remaining geometries come in two cases:

A product of two 2-dimensional geometries $$S^2\times\mathbb{E}^2$$ and $$S^2\times\mathbf{H}^2_{\mathbb{R}}$$.

A product of a 3-dimensional geometry with $$\mathbb{R}$$. These are $$S^3\times\mathbb{R}$$, $$\mathbf{H}^3_{\mathbb{R}}\times\mathbb{R}$$, and $$\widetilde{\mathrm{SL}_2(\mathbb{R})}\times\mathbb{R}$$.