Polygonal presentations: why no two-letter words?

Question

In Lee's book Introduction to Topological Manifolds, he discusses polygonal presentations of surfaces. He does so by means of words $W_1, \dotsc, W_n$ such that each letter that appears must appear exactly twice in $W_1, \dotsc, W_n$. He requires that words be at least three letters in length, except that two letters is allowed when there is only one word.

What kind of pathological behavior is he trying to avoid with this restriction? I'm not sure what can go wrong with multiple two-letter words.

Answer 1

The restriction is not meant to avoid pathological behavior; I just put it there for convenience. For example, it would be perfectly reasonable to interpret $\langle a,b\mid aa^{-1}, bb^{-1}\rangle$ as a presentation of a disjoint union of two spheres, or $\langle a,b\mid ab, ab^{-1}\rangle$ as a presentation of a projective plane. But we don't need these for proving the classification theorem, and allowing them would require a lot more special-case arguments in the proofs, because two-letter words don't correspond to polygons.

Answer 2

In a polygonal presentation, $\langle S | W_1, W_2, \cdots, W_n\rangle$, if any letter appears in any word $W_i$ more than twice (so that your word looks like $\cdots aaaa \cdots$, as an example), then in the geometric realization, you have to paste multiple consecutive edges in the same orientation, so that every neighborhood of any point on one of the edges labelled $a$ looks like multiple half-disks pasted together along the diameters. This is not homeomorphic to $\Bbb R^2$, so the corresponding geometric realization is not a surface and thus the polygonal presentation fails to be a surface presentation.

On the other hand, if only a single letter appears in the word, an open edge is left in the geometric realization, and any neighborhood of a point on that edge would be homeomorphic to $\Bbb R \times \Bbb R^+$, so that it may be a $2$-manifold with boundary, but not a $2$-manifold/surface.

Each word $W_i$ has to be of length at least $3$ because you want to have a corresponding convex polygon $P_i$ while constructing the geometric realization, and there is no polygon with smaller than $3$ sides. This is allowed if there is only one word to include spheres $\langle a | aa^{-1}\rangle$ and projective plane $\langle a | aa\rangle$ as geometric realizations.