Geometric Topology 10

Manufacturing Surfaces

We shall be concerned with connected compact surfaces initally without boundary. These come in two families: orientable and non-orientable. The former ones can be embedded into $\mathbb{R}^3$, the others not (unless they have a boundary like the Mobius band). We shall classify both types.

Definition: A surface without boundary (or $2$-manifold) is a Hausdorff topological space, each point of which lies in an open set that is homeomorphic to a disk in $\mathbb{R}^2$.

Examples:

• A torus
• A 'fat' trefoil knot

Cartesian Equations

Exercise: The equations all describe subsets of $\mathbb{R}^3$. Which are surfaces? Which are connected and which are compact? What do these look like in space?

1. $|x| + |y|+|z|=1$,
2. $yz+zx+xy=0$,
3. $xyz=1$,
4. $x^8+y^8+z^8=1000$,
5. $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$

Models for Surfaces

If enough cuts are made on a surface, so it is subdivided by a graph of vertices and edges, the pieces can be deformed and laid out as plane figures. The surface can then be reconstructed by 'sewing' along the cuts. In four simple cases, a single square suffices with the cuts defining pairs of edges.

We can define the models of four surfaces as follows:

1. Torus: $aba^{-1}b^{-1}$
2. Klein Bottle: $aba^{-1}b$
3. Projective Plane: $abab$
4. Sphere: $abb^-1a^-1

By labelling the edges, and reading clockwise from the top left, these models can be described by the 'words' above. The last two can be simplified $AA$ and $AA^{-1}$.

Embedded Surfaces in $\mathbb{R}^3$

A compact surface $\mathscr{M}$ is embedded in $\mathbb{R}^3$ if there is a continuous injective map $\mathscr{M} \rightarrow \mathbb{R}^3$. It will then be homeomorphic to its image.

As a topological space, the unit sphere:

is what we get when we take a blue disk, push it into a bag and fasten its top boundary up with a 'zip'.

The symbol $S$ stands for any surface homeomorphic to $S^2(1)$, for example the octahedron $|x|+|y|+|z|=1$. We have already seen the torus parametrized by a rectangle $\mathscr{R}$ (equivalently square). All $4$ vertices of $\mathscr{R}$ map to common point $p$ on the torus, and the $4$ edges of $mathscr{R}$ are identified in pairs and mapped to $2$ circles intersecting in $p$.

Immersed Surfaces in $\mathbb{R}^3$

• The Klein bottle $K$ is formed by passing a cylinder through itself. It can be mapped almost injectively into $\mathbb{R}^3$, but there is a circle of self-intersection that actually represents two disjoint circles on $K$.
• The projective plane $P$ can be immersed in $\mathbb{R}^3$ as a sphere with crosscap, parametrized by $\{(\cos u \sin 2v, \sin u \sin 2v, \cos^2 v - \cos^2 u \sin^2 v)\}$. It has a segment of self-intersection, where the image passes through itself.

These immersed surfaces in $\mathbb{R}^3$ are one-sided in the sense that the image of a continuous path in $P$ or $K$ can travel to all sides.

Classification

Our next aim is to prove this main theorem:

Main theorem (v1): Any connected compact surface without boundary is homeomorphic to one of the following:

1. A sphere $S$, or a sphere with $g \geq 1$ handles, equivalently a torus with $g$ holes.
2. A sphere with $h \geq 1$ crosscaps.

In case 1, the surface $\mathscr{M}$ can be embedded in $\mathbb{R}^3$ and is orientable. The integer $g$ is called its genus, and $S$ itself corresponds to $g=0$. One can regard a surface of genus $g$ as the 'connected sum' (in analogy of knots) of $g$ tori, and one often writes $\mathscr{M} = \overbrace{T \# ... \# T}^g = gT$, where "$=$" really means "homeomorphic to".

In (2), the surface must contain a Mobius band and is non-orientable. Any such surface can be regarded as the connected sum of $h$ copies of the projective plane, so $\mathscr{M} = hP$. We shall see that the Klein bottle has $h=2$. so we write $K = P \# P = 2P$. A key step in the proof of the main theorem is to show that $K \# P = T \# P$.