# Do all combinatorially distinct fundamental polygons correspond to surfaces?

The topology of a closed surface can be constructed by identifying edges of a fundamental polygon of an even number $2n$ of edges. Labeling the edges and using $\pm 1$ exponents to indicate direction, the construction can be specified by a string of $2n$ symbols: $a b a^{-1} b^{-1}$ for the torus, $a a b b$ for the Klein bottle, etc. (Reflecting a comment by Benjamin Steinberg:) Each letter in a fundamental polygon string appears exactly twice.

Let's say two symbol strings are equivalent if they are related by a combination of (a) circular permutation, (b) reflection/reversal, or (c) symbol permutation/relabeling. For $n=1$, there are two distinct strings, $aa$ and $aa^{-1}$. For $n=2$, I think (not certain) these are the combinatorially distinct strings: $$aabb,\; aa^{-1}bb,\; a^{-1}abb,\; aa^{-1}bb^{-1},\; a^{-1}abb^{-1},\;$$ $$abab \;, aba^{-1}b,\; aba^{-1}b^{-1},\; a^{-1}ba^{-1}b$$

Two questions:

Q1. Does every possible such string correspond to some surface?

Q2. Might two combinatorially distinct strings correspond to the same surface?

Perhaps $aa^{-1}bb^{-1}$ and $a^{-1}abb^{-1}$ describe the same surface, as they only differ in $aa^{-1}$ vs. $a^{-1}a$?

• Doesn't permutation of $a$ and $a^{-1}$ leave the surface invariant? Or am I saying something really stupid? Jun 27 '14 at 12:16
• More likely abab and a'ba'b are the same surface. (Not feeling up to superscripts right now.) Jun 27 '14 at 12:22
• I think that any identification of this form corresponds to a triangulated surface. E.g. in the case of the square, take the 3x3 grid and subdivide each square in two triangles. Every side is adjacent to two triangles, and every vertex has link homeomorphic to $S^1$. Now it should be a matter of counting: there are only so many surfaces you can represent with a $2g$-gon (since $\chi$ is bounded by a linear function of $g$). Jun 27 '14 at 12:26
• aabb is nonorientable; also, it has one vertex, two edges, and one face, so Euler characteristic 0; the same is true for aba'b, so they are homeomorphic (in fact, each is a Klein bottle). This sort of analysis can be used to determine all of the diagrams. Jun 27 '14 at 12:44

1. Yes. 2. Yes. (I suppose that the surfaces are "the same" if they are homeomorphic).

For 1, it is sufficient to check the definition of surface: that every point has a neigborhood homeomorphic to the disc. For interior points of the polygon, and for points on the sides, this is evident, and for the corners this is easy.

For 2, just recall classification of all possible compact surfaces up to homeomorphism. There is one integer invariant, the Euler characteristic, and the 2-valued invariant, orientability. The Euler characteristic is easily computed from your word: if you have 2n edges of your polygon, they will give $n$ edges after gluing, and suppose that you obtain $v$ vertices after gluing. Then the Euler characteristic is $1-n+v$ which is between $2-n$ and $2$. And orientability has two values. So you have at most $2n$ topologically different surfaces from strings of length $2n$.

And you see that for given length there are much more classes of words than $2n$.

• Isn't this a proof that the answer to 2. is Yes, i.e. that combinatorially distinct strings can correspond to the same surface ? Jun 27 '14 at 12:58
• Arnaud, thanks for your correction. Jun 27 '14 at 13:09
• Also, you need to do orientable and unorientable surfaces separately. Jun 27 '14 at 13:10
• I am confused. There are one-relator groups with even length relations that are not surface groups. The string $ab^6a^{-1}b^{-8}$ defines a non-Hopfian Baumslag-Solitar group and hence is not a surface group. Jun 27 '14 at 14:20
• To get a surface each letter should appear exactly twice. Maybe this was implicit in the question. Jun 27 '14 at 17:31

Great question!

There's an interesting comment at https://math.stackexchange.com/questions/204405/for-an-n-gon-how-many-fundamental-polygons-are-there

"we ... disallow discontinuities, [i.e] two A edges to be adjacent with "wrong" directions." In the first 3 polygons of the top row, the head of A or B is attached to its own tail so these are ignored.

Another insight is that the orientation of the arrows is a guide for attachment and is invariant if direction is inverted for all identical edges, say flip all A edges.

So the last two polygons on the top row give the same glueing for a sphere, just flip the direction of the red lines to turn one polygon in to the other.
On the bottom row you have: Real Project Plane, Klein Bottle, Torus and Real Project Plane again (invert the red lines to equal the 1st RPP).