Manifolds and Surfaces

A manifold is a space that looks, near each of its points, like ordinary Euclidean space: a surface is locally a piece of the plane, a three-manifold locally a piece of three-dimensional space. The universe, if it is finite, is presumably a three-manifold; the configuration space of a rigid body is the three-manifold of rotations; the Poincaré conjecture is about closed three-manifolds. This chapter defines manifolds precisely and then solves completely the problem that the Poincaré conjecture poses one dimension up: which closed surfaces are there?

The answer, the classification of surfaces, is a list: the sphere, the torus, the surfaces with more holes, and their non-orientable cousins. Each is determined by two pieces of information, orientability and one integer, the Euler characteristic. The integer is the same one that forces a football to have exactly twelve pentagons. In the list, only the sphere has every loop shrinkable to a point: the two-dimensional Poincaré theorem, proved by classification.

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

• define topological manifolds, with and without boundary, and say why each condition in the definition is there;
• describe surfaces by gluing polygons, and compute their Euler characteristics;
• form connected sums, and recognise orientable and non-orientable surfaces;
• state the classification of compact surfaces and sketch its proof;
• prove that a trivalent polyhedron made of pentagons and hexagons has exactly twelve pentagons.

3.1Twelve pentagons

3.2Manifolds

A homeomorphism from an open set $U \subseteq M$ onto an open subset of $\mathbb{R}^n$ is a chart. The dimension is well defined: an open subset of $\mathbb{R}^m$ is never homeomorphic to an open subset of $\mathbb{R}^n$ for $m \neq n$ (Brouwer's invariance of domain, 1912; proved with homology or degree theory, 7A.7 Smooth Topology, 7A.8 Homology in Brief). The two extra conditions exclude the line with two origins (not Hausdorff) and spaces too big to carry a metric (not second countable).

Examples. $\mathbb{R}^n$ and its open subsets. The sphere $S^n$: the stereographic projections from the north and south poles are two charts that cover it (7A.2 Compactness and Compactification). The torus $T^n = \mathbb{R}^n/\mathbb{Z}^n$ and the projective space $\mathbb{R}P^n = S^n/\{\pm1\}$, where small balls in the covering space map homeomorphically to the quotient (7A.6 Covering Spaces). Products of manifolds, such as $S^2\times S^1$. Graphs of continuous functions $\mathbb{R}^n \to \mathbb{R}^k$. The group $SO(3)$ of rotations, a three-manifold (7A.6 Covering Spaces).

A manifold with boundary is defined the same way with charts to open subsets of the closed half-space $\{x_n \geq 0\}$; its boundary points are those sent to $\{x_n = 0\}$, and the boundary $\partial M$ is a manifold of dimension $n - 1$ without boundary. The closed disc $D^n$ has boundary $S^{n-1}$; the Möbius band has boundary a single circle. A closed manifold is one that is compact and has no boundary. The Poincaré conjecture is about closed three-manifolds.

3.3Building surfaces from polygons

A surface can be described by gluing the edges of polygons in pairs, as in 7A.1 Topological Spaces and Quotients. Label the edges of a polygon with letters, going around the boundary; a letter appearing twice means those edges are glued, matching their directions, and $a^{-1}$ means the edge $a$ traversed backwards. Then:

word	surface
$aa^{-1}$	sphere $S^2$
$aba^{-1}b^{-1}$	torus $T^2$
$aa$	projective plane $\mathbb{R}P^2$
$abab^{-1}$	Klein bottle
$a_1b_1a_1^{-1}b_1^{-1}a_2b_2a_2^{-1}b_2^{-1}$	genus-2 surface

The genus-2 octagon was drawn in the hyperbolic disc in 5A.5 Uniformization and the Two-Dimensional Ricci Flow.

More generally, a cell complex is built by attaching discs of increasing dimension along their boundaries: points (0-cells), then edges (1-cells) attached at their endpoints, then polygons (2-cells) attached along loops of edges, and so on (Lee, chapter 5). A triangulation of a surface is a decomposition into triangles meeting edge to edge. Tibor Radó proved in 1925 that every surface can be triangulated, and Edwin Moise proved the same for three-manifolds in 1952 (8A.1 Smooth Structures).

It does not depend on the decomposition, a fact best proved with homology (7A.8 Homology in Brief); here it can be checked on examples. For the sphere as a polygon with word $aa^{-1}$: after gluing there are $2$ vertices, $1$ edge and $1$ face, so $\chi = 2$; the surface of a cube gives $8 - 12 + 6 = 2$ too. For the torus from the square $aba^{-1}b^{-1}$: all four corners become $1$ vertex, the edges become $2$, and there is $1$ face, so $\chi = 0$. For $\mathbb{R}P^2$ from $aa$: $\chi = 1 - 1 + 1 = 1$.

Connected sum. Given two connected surfaces, remove a small open disc from each and glue the two boundary circles together: the result is the connected sum $\Sigma_1\#\Sigma_2$ (Figure 3.2). Since removing a disc lowers $\chi$ by $1$ (one face less) and gluing along a circle doesn't change it (a circle has $\chi = 0$),

Orientability. A surface is orientable if it contains no Möbius band, equivalently if a consistent choice of "clockwise" can be made over the whole surface. The sphere and torus are orientable; $\mathbb{R}P^2$ and the Klein bottle contain Möbius bands and are not.

3.4The classification of surfaces

The relation $\mathbb{R}P^2\#T^2 \cong \mathbb{R}P^2\#\mathbb{R}P^2\#\mathbb{R}P^2$ shows that the connected sum does not have unique "cancellation" when non-orientable surfaces are involved; for orientable surfaces the genus is all there is. The Klein bottle is $N_2 = \mathbb{R}P^2\#\mathbb{R}P^2$.

The two-dimensional Poincaré theorem. In 7A.5 Computing π₁ the fundamental group of each surface in the list is computed: it is trivial for $S^2$ and non-trivial for every other surface (for instance $\mathbb{Z}^2$ for the torus). So a closed surface in which every loop can be shrunk to a point is a sphere. The proof is by classification: list everything, then check. In three dimensions no such list was available, and the Poincaré conjecture asked whether the same conclusion holds anyway. Perelman's proof does produce a kind of classification, through geometrization (10A.5 Thurston’s Eight Geometries, 12C.4 Geometrization).

3.5History

René Descartes found the angle-defect form of the polyhedron formula around 1630, in notes published only in 1860; Leonhard Euler stated $V - E + F = 2$ in a letter of 1750 and published it in the following years. The classification of orientable surfaces was obtained by August Möbius (1863) and Camille Jordan (1866), and the non-orientable case was added later; the first complete rigorous proof, based on triangulations, was given by Max Dehn and Poul Heegaard in 1907 and in a polished combinatorial form by Henry Brahana in 1921. Radó's triangulation theorem dates from 1925 and Brouwer's invariance of domain from 1912. Buckminsterfullerene was discovered in 1985.

3.6Exercises

Solution

$V = \frac23E$, $F = p + h$, and $E = \frac{5p + 6h}{2}$. Then $V - E + F = -\frac13E + p + h = -\frac{5p + 6h}{6} + p + h = \frac p6 = 2$, so $p = 12$. With $s$ heptagons, the same computation gives $\frac p6 - \frac s6 = 2$, so $p = 12 + s$: each heptagon needs an extra pentagon (as in some curved carbon structures).

Solution

All four corners are identified ($1$ vertex), $2$ edges, $1$ face: $\chi = 0 = 1 + 1 - 2$. Not homeomorphic: the Klein bottle is non-orientable. Euler characteristic alone does not distinguish orientable from non-orientable surfaces; the theorem needs both invariants.

Solution

Cube $8 - 12 + 6 = 2$; octahedron $6 - 12 + 8 = 2$; icosahedron $12 - 30 + 20 = 2$. The picture frame's surface is a torus, and any decomposition of it gives $0$ (for example, with quadrilateral faces: $16 - 32 + 16 = 0$).

Solution

(a) Defect per vertex $360° - 108° - 240° = 12°$, total $720°$. (b) The total angle is $\sum_{\text{faces}}(k_f - 2)\pi = (2E - 2F)\pi$. The total defect is $2\pi V - (2E - 2F)\pi = 2\pi(V - E + F) = 2\pi\chi$.