Geometric Topology 15

Final Classification

We shall now generalise the classification theorem to connected compact surfaces ( $2$ -manifolds) with boundary. Such a surface can be defined by asserting that every point is contained in a subset that is homeomorphic to a closed disk in $\mathbb{R}^2$ , as this allows boundary points.

This matches the definition of a triangulation defined previously, and we can always regard such a surface $\mathscr{M}$ with boundary as a quotient $\hat{\mathscr{P}}$ of a single polygon whose own boundary $\partial \mathscr{P}$ is defined by a word $W$ in which each letter appears twice or once.

If the boundary is empty, our previous theorem asserts that we can convert $W$ into one of the normal forms:

\mathbb{A}_g = (a_1b_1a_1^{-1}b_1^{-1}) \cdots (a_gb_g a_g^{-1}b_g^{-1})

for $g \geq 1$ , with $\mathbb{A}_0 = aa^{-1}$

Or:

\mathbb{C}_g = (c_1c_1)\cdots (c_hc_h)

for $h \geq 1$ .

To allow for a boundary, it turns out we merely need to add one or more cuffs with groups of letters:

\mathbb{D}_r = (u_1d_1u_1^{-1}) \cdots (u_rd_ru_r^{-1})

for $r \geq 1$ , with $\mathbb{D}_0 = \varnothing$ .

The exact choice of letters is immaterial provided they do not clash.

Topological Type

Main Theorem (v3): Any connected compact surface $\mathscr{M}$ with boundary (possibly empty) arises from a polygon and a single word of exactly one of the following types:

$\mathbb{A}_0$ or $\mathbb{D}_r$ with $r \geq 1$ , or $\mathbb{A}_g \mathbb{D}_r$ with $g \geq 1$ and $r \geq 0$ .
$\mathbb{C}_h \mathbb{D}_r$ with $h \geq 1$ and $r \geq 0$ .

The integer $r$ represents the number of boundary components. Since the removal of a disk (homeomorphic to the interior of a triangle) affects only $F$ , it reduces $\chi$ by $1$ . This can also be understood by tracing vertices and counting edges in the normal forms above:

$\chi (\mathbb{A}_g \mathbb{D}_r) = 2 - 2g - r$
$\chi (\mathbb{C}_h \mathbb{D}_r) = 2 - h - r$ .

Both $\chi$ and $r$ are topological invariants, meaning that homeomorphic surfaces have the same values. In case (1), $g$ is again called the genus of the orientable surface. We have now justified the preview we initially gave of this theorem.

Corollary: The topological type (homeomorphism class) of a connected compact surface is entirely determined by its orientability, together with $\chi$ and $r$ .

A Trickier Example

We previously showed that:

abda^{-1} eb^{-1} \sim xyx^{-1}y^{-1}ed \sim (xyx^{-1}y^{-1})(ufu^{-1}) \sim \mathbb{A}_1 \mathbb{D}_1.

The left-hand word encodes the insertion of distinct edges in the two bottom corners of the square defining the torus $T$ . Let's see what happens when we replace $T$ with $P$ :

Example: The boundary code $W = abdaeb$ determines a non-orientable surface with boundary. Since $\chi = 2-4+1 = -1$ , it could be $\mathbb{C}_1 \mathbb{D}_2$ or $\mathbb{C}_2 \mathbb{D}_1$ .

W \sim \boxed{{\color{red}{a}}bd}aeb \ (x = abd \implies a = xd^{-1}b^{-1}) \\ \sim xxd^{-1}(b^{-1}eb).

We can see pictorially that the naked edge $d$ cannot be combined with the edge $e$ to get a common boundary, so $r=2$ and the normal form is $\mathbb{C}_1\mathbb{D}_2$ .

Cuffs occur naturally in an attempt to convert a word into normal form. But one may also be left with one or more naked letters, like $d^{-1}$ above. One can convert them into cuffs using the trick we used previously, which shows how a single letter $u$ can be 'passed through' $xx \sim \mathbb{C}_1$ . The next lemma establishes the analogous result when $\mathbb{C}_1$ is replaced by $\mathbb{A}_1$ .

Commutation Relations

Lemma A: Represent $\mathbb{A}_1$ by $aba^{-1}b^{-1}$ . Let $W = u\mathbb{A}_1 E$ be a word, where $E$ is any group of letters not involving $u,a,b$ . Then $W = \mathbb{A}_1 uE$ , using the operations we defined previously.

Proof: This is accomplished by means of four (cut and paste) substitutions, each of which moves $u$ into a different position within the group $\mathbb{A}_1$ (in the order $1,4,3,2,5$ ).

Note that $E$ merely acts as a buffer in the proof:

u\mathbb{A}_1 E \sim \boxed{u\color{red}{a}} ba^{-1}b^{-1}E \ (x = ua \implies a^{-1} = x^{-1}u) \\ \sim \boxed{x\color{red}{b}}x^{-1}ub^{-1}E \ (y = xb \implies b^{-1} = y^{-1}x) \\ \sim \boxed{ycolor{red}{x^{-1}}}uy^{-1}xE \ (z = yx^{-1} \implies x = z^{-1}y) \\ \sim \boxed{zu \color{red}{y^{-1}}} z^{-1} y E \ (w = zuy^{-1} \implies y = w^{-1} zu) \\ \sim (wz^{-1}w^{-1}z) uE \sim \mathbb{A}_1 uE. \ \blacksquare

Together with the easier result that $u \mathbb{C}_1 E \sim \mathbb{C}_1 u E$ , Lemma A is an ingredient in completing the proof of theorem v3. For, if we are left with a word of the form $W = \mathbb{A}_g d$ , we can deduce that $W \sim \mathbb{A}_g du^{-1}u \sim u \mathbb{A}_g du^{-1} \sim \mathbb{A}_g \mathbb{D}_1$ .

Genus of a Knot

Definition: The genus $g(K)$ of a knot $K$ is the least genus of any orientable surface that it bounds. It is therefore a knot invariant.

If $g(K) = 0$ then $K$ bounds a disk and is the unknot. It is also known that:

$g(K)$ is also achieved by Seifert's algorithm if the knot is alternating, but not in general.
The genus of a connected sum is additive: $g(K_1 \# K_2) = g(K_1) + g(K_2)$ . The connected sum requires one to first select an orientation for each knot, and in general its ambient isotopy class will depend on this choice.

Example: The Seifert surface $\mathscr{S}$ of the Whitehead link has:

\chi(\mathscr{S}) = |\sigma | - c = 3 - 5 = - 2 \\ \implies -2 = 2-2g - r = -2g \\ \implies g = 1.

Therefore $\mathscr{S}$ is homeomorphic to a torus minus two disjoint disks, though this may not be obvious from its picture!

Connected Sum of Surfaces

We know that any connected compact surface without boundary is a sphere with either $g \geq 0$ handles or $h \geq 1$ crosscaps. This can be made precise using the concept of connected sum. The latter is easiest to define combinatorially in the first instance.

Definition: Let $\mathscr{M}_1, \mathscr{M}_2$ be two surfaces without boundary described as quotients of polygons by words $W_1,W_2$ without letters in common. Their connected sum is the surface (denoted by $\mathscr{M}_1 \# \mathscr{M}_2$ ) defined by $W_1W_2$ , i.e. the operation of juxtaposition.

To regard $W_1W_2$ as the boundary of a new polygon, we need to 'cut' the boundary of $\mathscr{P}_1$ where $W_1$ starts/ends and insert $W_2$ (also cut from $\mathscr{P}_2$ ). For the definition to make sense, we should be able to insert $W_2$ between any two edges of $W_1$ . Indeed, the surface defined by $W_1W_2$ must only depend on those defined by $\mathscr{M}_1$ and $\mathscr{M}_2$ :

Lemma J: Suppose that $W_1,W_2$ have even length and no unpaired letters, and $W_1', W_2'$ similarly. If $W_i \sim W_i'$ for $i=1,2$ then $W_1W_2 \sim W_1' W_2'$ .

Recall that $W_i \sim W_i'$ means that one can be obtained from the other by a sequence of operations we defined previously, and the quotients $\hat{\mathscr{P}}_i$ and $\hat{\mathscr{P}}_i'$ are then homeomorphic.

Rearranging Independent Words

Lemma J provides a shortcut to dreaming up a sequence of valid operations on words to produce homeomorphic surfaces. We shall illustrate it with examples in place of a proof. Note that $W_1,W_2$ must be 'independent' (meaning no letters in common).

Examples:

Take $W_1 = W_1'$ ( $= aba^{-1}b^{-1}$ , for example), $W_2 = uccu^{-1}$ and $W_2' = cc$ . We know that $W_2 \sim W_2'$ by rotation, so the theorem implies: $W_1 uccu^{-1} \sim W_1 cc.$

We can deduce this by a (cut and paste) substitution, inverting what we did in a previous example:

W_1 W_2 = W_1 \boxed{uc}cu^{-1}, \ x = uc, c =u^{-1}x \\ \sim W_1 xu^{-1}xu^{-1} \sim W_1 W_2'.

Lemma J implies that: $W_1 uyx^{-1}y^{-1}u^{-1} \sim W_1 xyx^{-1}y^{-1},$

which is also a consequence of lemma A. One might encounter the left-hand word during conversion to normal form.

Geometrical Interpretation

Given a word $W = W_1W_2$ not containing the letter $x$ , we have:

W \sim W_1 x^{-1} + xW_2.

Each summand represents (taken alone) a surface minus a disk. The valid pasting operation identifies their boundaries circles $x$ , so they 'stick together' as a single connected surface. Lemma J is a way of asserting that it does not matter from which parts of the respective surfaces the disks are removed.

Examples:

We can interpret the word $W_1 xx^{-1}$ as the connected sum $\mathscr{M}_1 \# S$ of $\mathscr{M}_1$ with a sphere, which is of course homeomorphic to $\mathscr{M}_1$ .
Take $W_1 = aba^{-1}b^{-1}$ and $W_2 = cc$ . A disk is removed from $T$ and one from $P$ , allowing the boundary circles to be glued to obtain $T \# P$ . We now know that:
$T \# P = P \# T = P \# P \# P = K \# P.$
Here $=$ denotes 'same homeomorphism class'.

Further Properties

For homeomorphism classes of surfaces, the connected sum operation $\#$ is associative (since $(W_1W_2)W_3 = W_1(W_2W_3)$ ). It is commutative since (since $W_1W_2 \sim W_2W_1$ ). Note that $xx^{-1}$ acts as the identity (since $Wxx^{-1} \sim W$ , and we are attaching a sphere).

Provided that $W_1$ has no single letters (so $\mathscr{M}_1$ has no boundary) it is true that $W_1x^{-1} \sim W_1 ux^{-1}u^{-1}$ , since adding the cuff does not affect the topology. This can be regarded as an extension of Lemma J. Similarly for $xW_2 \sim vxv^{-1}W_2$ , hence:

W_1W_2 \sim W_1 ux^{-1}u^{-1} + vxv^{-1}W_2 \sim u^{-1}W_1 uv^{-1}W_1 v \sim W_1 wW_2 w^{-1},

where $w = uv^{-1}$ . The last word represents a surface in which (if in space) $\mathscr{M}_1,\mathscr{M}_2$ are joined by a tube. The image shows two ways of visualising $T \# T$ .

Euler Characteristics

Proposition: $\chi (\mathscr{M}\_1 \# \mathscr{M}\_2) = \chi (\mathscr{M}\_1) + \chi (\mathscr{M}\_2) -2$ .

Proof: Compute $\chi$ by counting $E,V,F$ arising from $W_1W_2$ or by joining $\mathscr{M}_1, \mathscr{M}_2$ across a triangle whose interior is removed. $\blacksquare$

Example: Take a connected graph $G$ in space with $v$ vertices and $e$ edges. Place a sphere at each vertex, and attach a tube between spheres for each edge, so as to form an oriented surface $\mathscr{M}$ of genus $g$ . Unless the graph has no cycles (so is a tree), $\mathscr{M}$ cannot be a connected sum of spheres, since we will be adding a handle from a surface to itself at some point. Nevertheless, we can apply the proposition to deduce that:

2 - 2g = \chi = 2v - 2e \implies g = 1 - v + e.

If $G$ is a planar graph, and is drawn in the plane with $f-1$ internal regions, then it is clear that $\mathscr{M}$ is a sphere with $f-1$ handles, which is consistent with Euler's formula $v-e+f=2$ .