Geometric Topology 14

Surfaces Without Boundary

We shall continue to model surfaces combinatorially, by means of one or more words. But to allow boundaries we shall allow each letter (or edge) to appear twice or once.

Example: As mentioned previously, the word $abcb$ describes a Mobius band $M$ whose boundary $d = ac^{-1}$ is homeomorphic to a circle. That's because the start of $a$ and end of $c^{-1}$ get mapped to the same point of $M$, as do the end of $a$ and start of $c^{-1}$. If we set $ab = x^{-1}$ (equivalent to cutting $abx + x^{-1}cb$ and then pasting) then:

We can now 'sandwich' $d$, using an insert and two more (cut and paste) substitutions:

Cuffs

The trio $udu^{-1}$ within a word (in which $d,u$ do not appear elsewhere) is called a cuff. The letter $u$ helps to distinguish different boundary components, and the presence of a cuff indicates that one disk has been removed from the surface.

Proposition: If a word $W$ represents a surface $\mathscr{M} = \hat{\mathscr{P}}$ (with or without existing boundary), then $Wudu^{-1}$ represents a surface that is homeomorphic to $\mathscr{M}$ minus an open disk (assuming that $u,d$ do not appear in $W$).

Proof: A polygon with boundary $Wudu^{-1}$ is shown. Folding the $u$ edges defines the required homeomorphism between the quotients. $\blacksquare$

Naked Edges

A problem is that two separate single edges inserted into the boundary of a polygon do not always produce two components. Here they unite into a single curve:

Since all vertices of $x^{-1}axa^{-1} \sim \mathbb{A}_1$ are equivalent, $ed$ is homeomorphic to a circle, so the quotient surface is a torus with only one disk removed. To obtain two circles, one must 'insulate' the edges with cuffs; havind one this, the derivation above yields:

This word represents an orientable surface with $\chi = 3-6+1 = -2$, and each cuff contributes $-1$ (1 extra vertex and 2 edges).

The number $r$ of cuffs is the number of boundary components (necessarily connected compact $1$-manifolds, each homeomorphic to the circle). If the surface is orientable and embedded in $\mathbb{R}^3$, then the boundary is a link and each component is a knot. The latter will not in general be ambient isotopic to a circle (think Seifert surface).