Geometric Topology 11

Subdividing Surfaces

Given a surface (possibly with boundary), we cut it into smallish 'triangles' (possibly with curved sides) with matching directed edges, which can be labelled ($\xrightarrow{a}$, $\xrightarrow{b}$, etc.) so the same label only occurs on two adjacent triangles. Such a triangulation can always be chosen with the following properties:

1. Each edge belongs to at most two triangles
2. The set $st(v)$ ('star') of triangles sharing a vertex $v$ forms a polygon homeomorphic to a closed disk (this would also hold for a point on the boundary if there were one)
3. Two triangles are either disjoint or meet in one common edge or a single vertex.

Properties 1 and 2 guarantee the locally Euclidean property. Each edge must belong to exactly two triangles assuming (as currently) that there is no boundary.

Triangulating the Projective Plane

If we start with the rectangle or square model, we can divide it into $2$ triangles, but many more are needed to prevent the triangles from having two isolated vertices in common. From experimentation we can see that $18$ triangles suffice, but the minimum for $P$ is known to be $10$. Here the boundary is encoded by the word $abc def abc def$, equivalent to $ADAD$ or even $AA$. The right-handed set of axes can be propagated from one triangle to another, but when it passes across the boundary of the square it becomes left-handed. So there is no consistent sense of clockwise/anti-clockwise rotation, and the surface $\mathscr{M}$ is said to be non-orientable. This is like having a one-sided surface in $\mathbb{R}^3$, but the concept does not require one to visualise $\mathbb{M}$ in space.

Polygonal Surfaces

Once a surface $\mathscr{M}$ is triangulated, the triangular pieces can be separated and then partially re-assembled in the plane, adding one triangle at a time by pairing $2$ edges. When this process is complete, we can avoid isolated vertices because rule (2) implies that a punctured neighbourhood of $v$ is connected, so we can always find triangles filling in above and/or below $v$, with their edges identified.

The order of matchings can result in different configurations but, assuming $\mathscr{M}$ is connected we can always suppose that the final result is a single (filled) polygon. Its edges will then be labelled, like the squares previously.

In practice, one can relax the rules of triangulation, and divide the surface into a finite union of polygons whose boundary edges are matched in pairs. Each polygon has a boundary code consisting of a word $W$ (such as $abca^{-1}d$). The symbol $a^{-1}$ indicates that $a$ and $a^{-1}$ have an opposite sense of rotation around the polygon, though the inverse is omitted if the symbol accompanies an arrow.

A Hexagonal Model of a Torus

This $6$-sided polygon $\mathscr{P}$ is triangulated with edges labelled in accordance with the rows and columns (read upwards) of the matrix above. The boundary code:

determines a surface $\mathscr{M}$ which we shall explain is a topological quotient of $\mathscr{P}$.

Exercises:

1. No occurrence of $...x...x...$ means that $\mathscr{M}$ has a consistent sense of clockwise/anticlockwise rotation, so is orientable.
2. Move $3$ triangles so as to transform the hexagon into an $8$-sided polygon with boundary code $dh^{-1}g^{-1}fhd{-1}f^{-1}g = DGD^{-1}G^{-1}$, which is equivalent to that of a torus.

A Dehn Twist

We now know that the quotient of the hexagon with boundary code $a^{-1}ie^{-1}ai^{-1}e$ is homeomorphic to the torus. Consequently there must exist a continuous surjective mapping:

where $\mathscr{M}$ is a torus of revolution. The map $q$ can be understood by cutting teh torus along a circle, then twisting it by 180 degrees before sewing it back together. This technique was used by Max Dehn around 1910 to generate the so-called mapping class group of a surface.

Surfaces as Quotients

Given a connected surface $\mathscr{M}$ without boundary, recall that we can make a model of it as a polygon $\mathscr{P}$, with boundary $\partial \mathscr{P}$ consisting of $2n$ sides. A boundary code consists of a word with $2n$ letters occurring in pairs. The edges of each pair need to be 'sewn' or 'glued' together so as to eliminate the boundary of $\mathscr{P}$ and recover $\mathscr{M}$.

More formally, the code defines an equivalence relation on $\mathscr{P}$ whose classes have size $1,2$ or more. An interior point of $\mathscr{P}$ is only equivalent to itself. An interior point of an edge $a$ is equivalent to exactly one other point on the other edge labelled $a$ or $a^{-1}$. A vertex will be equivalent to at least one other point, unless it occurs in the middle of $aa^{-1}$ or $a^{-1}a$, which (if a $2$-gon on its own) represents the sphere $S$.

$\mathscr{M}$ can be defined as the set $\hat{\mathscr{P}}$ of equivalence classes, and there is a surjective mapping called the projection, $q : \mathscr{P} \rightarrow \mathscr{M}$. The image $\mathscr{M}$ is not merely a set, but a topological space: a subset $U$ of $\mathscr{M}$ is declared open if and only if $q^{-1}(U)$ is open in $\mathscr{P}$ (and so of the form $\mathscr{P} \cap V$ where $V$ is an open subset of $\mathbb{R}^2$). This makes $q$ continuous.

Example: Let $x \in \partial \mathscr{P}$ so that $m = q(x) = \{x,x'\}$. Then an open subset $q^{-1}(U)$ of $\mathscr{P}$ containing $x$ must contain a small semicircular region around both $x$ and $x'$.

Normal Form

Our task will be to understand when two different boundary codes (words) for $\mathscr{P}$ give rise to homeomorphic surfaces. For this purpose, we shall define a surface combinatorially, purely in terms of a word an the operations on it that preserve topological type. To interpret these operations, we shall rely on visual intuition to understand $\mathscr{M}$ as the quotient space $\hat{\mathscr{P}}$.

Main Theorem (v2): Any connected surface without boundary arises from a polygon with an even number of edges, identified in pairs, and is uniquely specified by exactly one of the following words:

1. $aa^{-1}$, or $a_1b_1a_1^{-1}b_1^{-1}...a_gb_ga_g^{-1}b_g^{-1}$ with $g \geq 1$,
2. $c_1c_1...c_hc_h$ with $h \geq 1$.

Don't forget that polygon means "filled polygon": an open interior together with boundary edges and vertices.

The two cases coincide with the previous discussed notions. In particular $a_ib_ia_i^{-1}b_i^{-1}$ signals the presence of an attached torus/handle, and $c_jc_j$ indicates the presence of a crosscap.

Sewing the Projective Plane

Consider the square model of the projective plane $P$ with boundary code $abab$. If we replace one of the $a$'s with $c$ we obtain the code $abcb$ that represents a surface with boundary. It is clear that $abcb$ is the code for describing a Mobius band $M$ in an abstract way. The boundary consists of $ac^{-1}$ which (in the view of matching vertices) is homeomorphic to a circle.

The simplest way of visualising $M$ in $\mathbb{R}^3$ is to regard the square as a piece of paper, and attach the two edges labelled $b$ by twisting the paper by 180 degrees. The boundary is then an unknot (whilst had the twist been 540 degrees it would have been a trefoil). Forgetting the band, we can flatten and deform the unknot into the boundary of a closed disk $D$, itself represented by a $2$-gon with code $ac^{-1}$. In notation that will be developed in the next sections, we can write:

which all goes to show that $P$ is obtained by uniting $D$ to $M$. One can try to visualise an immersion of $P$ in $\mathbb{R}^3$ by sewing the disk onto the twisted band, though in practice this requires some intersection of the disk with the band (so a self-intersection of $P$).

An Abstract Klein Bottle

We previously saw that the cylinder model of $K$ leads to a square with word $aba^{-1}b$, but this is not in normal form, so is not recognisable to the theorem. We can convert the word $aba^{-1}b$ into normal form with a "cut and paste" operation, namely we remove the triangle below the diagonal and re-position it so as to match up the two $b$ edges.

First we split the diagram into triangles $axb$ and $a^{-1}x^{-1}b$. These can be combined by "flipping" the second over to match the $b$ edges. In symbols:

where $\sim$ indicates that the associated surfaces are homeomorphic. The new word is $c_1c_1c_2c_2$ in the notation of the theorem, confirming that $K$ is a sphere with two crosscaps.