Fundamental polygons with infinite pairwise identifications

Question

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.

My question is: Does it make sense to have an infinite number of boundary identifications, i.e., does it define some topological object?

More specifically:

1. Does an infinite string of symbols representing pairwise identifications correspond to some surface? For example, the generalization of a non-orientable genus-$n$ surface as $n \rightarrow \infty$: $$a_1 a_1 a_2 a_2 a_3 a_3 \cdots a_i a_i \cdots \;.$$
2. Does it make sense to have an uncountable number of pairwise point identification around a circle? For example, parametrize the circle circumference from 0 to 1 and identify points with complementary binary representations: $$.011100100011\ldots \leftrightarrow .100011011100\ldots \;.$$

These extensions may be nonsensical, in which case I apologize for the distraction! But if something along these lines has been studied, I'd appreciate a reference. Thanks!

Edit. What was nonsensical was my bit-complement example, as pointed out by both Victor and Sergei. I'll leave it so their remarks make sense. I intended a more patternless pairing.

Answer 1

Identification spaces with infinitely complicated identifications do occur naturally in various contexts. Here are a few more remarks:

1. If you define an involution on a countable set of intervals on the boundary of a surface, as in Victor Protsak's remark, and extend the equivalence relation to finite equivalence classes of the closure (i.e., vertices), you can construct arbitrary noncompact surfaces. The classification of these is known, and rather interesting. In the orientable case: The first invariant is the set of ends of the space, meaning, the inverse limit of the set of components of complements of compact sets. I think this is the same as the closure of the equivalence relation extended to the rest of the circle, minus the surface, but I haven't thought it through properly. The set of ends can be an arbitrary closed homemorphism type of compact totally disconnected space. I believe it's known that the number of types has the cardinality of the continuum; it's easy to see it's at least $\omega$ and at most $2^\omega$. The second invariant is a closed subset of the set of ends, that is the limit of "handles", that is, pairs of curves that intersect in one point. Finally, if there are only finitely many handles, you need the genus.

2. There is a structure called a geodesic lamination in the disk: a closed collection of disjoint hyperbolic lines ending at infinity. Given a lamination, you can identify the endpoints. This kind of identification arises in multiple ways. First, it describes the structure of the cut locus" for a curve in the plane. For which pairs of points is there a circle whose interior is contained on the inside, but touches only those two points? Second, if you identify all lines to points, the resulting space is still homeomorphic to the plane. They give explanations for the structure of Julia sets and limit sets of Kleinian groups, etc. More generally, one can represent identifications by collections of lines that are allowed to cross, but the theory becomes more complicated. In the case of Julia sets, the identification is in some sense isometric. (Actually, the involution is nontrivial only on a set of measure 0 with dense complement, but it is a limit of isometric involution on finite collections of intervals.) If you identify the circle using two geodesic laminations, one on the inside and one on the outside, you typically get $S^2$, and the circle becomes a space-filing curve: a sort of Hamiltonian path that is a limit of simple curves.

3. For a dynamical system, it is natural to look at the set of all orbits. This is an identification space that is somtimes "nice" but more often is (in Ivanov's words) weird: non-Hausdorff and failing other separation properties. Nonetheless, these quotient spaces often carry structure that is helpful to understanding the dynamics. A simple example: the flow in the plane minus the origin $\phi_t(x,y) = (\exp(t) x, \exp(-t) y) of a linear differential equation. The orbit space is a 1-dimensional manifold, because you can take a cross-section near any point that meets any orbit at most once, so it embeds in the quotient space. However, the quotient is not Hausdorff.

Answer 2

I am not sure what the question is, but will try to answer anyway. This is basic general topology stuff, sorry if you meant something deeper in your question.

For any topological space $X$ and any equivalence relation $\sim$ on $X$ there is a natural topology on the quotient $X/\sim$, see e.g. this Wikipedia article. Intuitively, the quotient space is the result of "gluing together" all points in every equivalence class of $\sim$.

Making a surface from a polygon is a partial case. For example, the gluing scheme $aabb$ represent the following equivalence relation on a square $ABCD$: an interior points is not equivalent to anything but itself, a point $X$ on the side $AB$ is equivalent to a point $X'$ in $BC$ such that $AX:XB=BX:XC$ and similarly for the sides $CD$ and $DA$. By transitivity, all four vertices $A,B,C,D$ are in the same equivalence class. (In other gluing schemes, such as $abab$, the vertices can form more than one equivalence class.) The quotient space is a Klein bottle. In general, it is easy to prove that for any gluing scheme of a $2n$-gon (where every letter appears exactly two times) the resulting quotient space is a compact surface without boundary. Then there is a topological classification of such surfaces, and so on.

Note that you don't have to begin with a polygon. Any shape homeomorphic to a polygon (e.g., a 2-dimensional round disc) will work the same way because homeomorphic spaces are identical from the topological perspective.

Speaking about generalizations, once you have defined a topological space that serves as a "generalized polygon" and an equivalence relation (the gluing rule) on it, you get a valid topological space as a quotient. However it can be some "weird" space rather than a surface, and it may depend on how exactly you generalize the notion of a polygon.

For example, if you asked about a two-sided infinite string like "$\dots a_{-1}a_{-1}a_0a_0a_1a_1\dots$", it would be natural to take the half-plane as a "polygon", divide its boundary into unit segments and glue them according to the string. Sometimes the result is a (non-compact) surface, sometimes it is not. Namely, if every equivalence class is finite, the quotient is a surface (= two-dimensional manifold), otherwise it is not (because it is not locally compact). In my example, all "vertices" end up in the same equivalence class, so the quotient is not a surface.

Alternatively, you could pack these infinitely many segments around a circle. In this case the quotient is compact but most likely it will fail to be a surface around accumulation points.

You second example is clear. You identify points $x$ and $y$ if $x+y=1$. Equivalently, you could divide the circle into two segments $[0,1/2]$ and $[1/2,1]$ and apply the gluing scheme $aa^{-1}$ to this "2-gon". The resulting surface is the sphere. Another well-known construclion is to identify every point on the circle with the opposite one - this is the same as "$aa$" gluing and the result is the projective plane. However if you make some weird pairwise identification, most likely the result will be some weird topological space.