Walking in Circles: Fundamental Groups

In algebraic topology, one of the most basic ideas is *the fundamental group* of a point in the space. The fundamental group tells you a lot about the basic structure or shape of the group in a reasonably simple way. The easiest way to understand the fundamental group is to think of it as the answer to the question: “What kinds of different ways can I circle around part of the space?”

To really understand what the fundamental group is, we need to think back to what I said about group
theory. Group theory is most basically about defining *symmetry*; and symmetry in turn means a kind of *immunity to transformation.* So given a point in a topological space, what does that mean? What is a transformation, and what would immunity mean?
In topology, almost everything comes back to continuous functions. To transform a space we use homeomorphisms. To transform a *point* within the space, we still want to use a continuous
function, but we don’t want one that maps to a different space. What kind of continuous function would make sense for a point?
The answer to that is to think about what a homeomorphism *means*, and informally try to
think of what the same kind of thing would mean for a point. Homeomorphism, informally, means that you can take a space, squish it, stretch it, pull it, twist it, and otherwise deform it *without* breaking it. The way that I’ve always thought of it, while not really entirely correct, gives a nice sense: you have a clay model of the space floating in a space with one more dimension. You can do anything you want with that ball of clay – as long as you don’t ever break any of its edges. As long as you do that, when you’re done, topologically speaking, you’ve got the same thing you started out with.
So think about that idea – of being able to squish and push things about within an embedding space, as long as we don’t break anything, and ending up with what we started. What would that mean in terms of a *point*? Starting from a point, I can *walk around* the space it’s part of in any way I want – so long as I don’t *jump* in any way that breaks my path – and if I wind up back where I started, then I haven’t really changed anything. That’s basically what the objects in the fundamental group of a point are: the *continuous loops* that follow some path through the space, both starting and ending at the point. Of course, in typical topological fashion, we don’t *care* if there are two paths that are basically the same, except that one is sort of an irregular oval, and the other is a circle. So the fundamental group isn’t really *all* of the loops; it’s a set of continuous loops such that *every continuous loop is a continuous deformation* of some object in the group. Groups also need some operation; the operation for loops in a topological space is basically concatenation: connect the end point of each loop to the start point of the other – which is itself another loop.
So that’s a pretty good *informal* version. Now let’s try to get down to the actual formal topological definition. We’re going to use a pretty standard topological trick: if we want to show that there’s a smooth path or a smooth transformation, we do it using a continuous function from the interval [0,1] to the path. So for a loop for point *p* (also called a loop *with base point p*), what we’ll do is create a continuous function *f* from [0,1] to the path of the loop. We can make it be a loop by saying that *f*(0)=*f*(1)=p. Putting that together: Given a topological space **T**, and a point *p* ∈ **T**, a *loop* is given by a continuous function *f : [0,1] → **T***, where *f(0)=p* and *f(1)=p*.
So now we have a loop. But up above we said that if you had two loops both with the same base point *p*, and you could transform one into the other *without* breaking, then you treated them as the same. How do we say that? Easy – we pull out exactly the same trick of using the interval. So the loops *f* : [0,1] → **T** and *g* : [0,1] → **T** with base point *p* are *equivalent* loops if and only if there’s a third continuous function *h* : [0,1] × [0,1] → **T**, such that:
1. ∀ x ∈ [0,1] : h(x,0) = f(x)
2. ∀ x ∈ [0,1] : h(x,1) = g(x)
3. ∀ x ∈ [0,1] : h(0,x)=h(1,x)=p
That’s all a rather fancy way of saying that that *f* and *g* can be continuously transformed
into each other: that there’s an infinite series of loops between *f* and *g*, each of which is minutely, infinitessimally different from the one before it, which show the smooth, continuous deformation from *f* to *g*. An easy way to think of it is that that series of loops is an *animation* of the deformation; each loop in the series is one frame; put them all together, and you’ll have a perfectly smooth animation of the deformation of *f* into *g*; run it backwards and you’ve got the deformation of *g* to *f*.
The function *h* that describes the mutual-transformability property of *f* and *g* is called a *homotopy* from *f* to *g*. Homotopies define a set of equivalence classes over loops with a common base point called *homotopy classes*.
Here’s where we come across a really neat trick. Suppose we’ve got a really pretty topological space. It’s connected – and in fact, not just connected, but [*path-connected*.][path-connect]. Then using the same kind of trick for defining equivalences, we can show that in fact, *all* points in the space have *the same* fundamental group. In that case, we call it the fundamental group not just of the point, but of the entire topological space.
To give you one neat example of a fundamental group. Suppose you’re looking at the topological space consisting of a circle. The loops inside of a circle are circles – one full pass around the circle, two passes, three passes, etc. If you think about that, and the idea of the group operation being concatenation of loops, you can hopefully see that the the fundamental group of the circle is equivalent to the group consisting of *(**Z**,+)*, that is, the group of integers with the addition operation. Each *clockwise* loop around the circle is (+1); each *counter-clockwise* loop is (-1). Concatenating the loops ends up accomplishing the same thing as adding the corresponding integers.
Neat, huh?
[path-connect]: http://scienceblogs.com/goodmath/2006/09/connectedness.php

0 thoughts on “Walking in Circles: Fundamental Groups

  1. John Armstrong

    Of course then we can consider the fundamental groupoid of a space X. This is a category whose objects are the points of X and whose morphisms from x to y are the paths from x to y up to homotopy. It’s a groupoid because every morphism from x to y has an inverse (up to homotopy) from y to x: just walk the path backwards.
    So the fundamental group with basepoint x is hom(x,x). Given another point y and a path p from y to x we can conjugate in the groupoid and send g in hom(x,x) to p-1gp in hom(y,y), which gives the isomorphisms you mention for a path-connected space. Of course the isomorphism isn’t canonical, but depends on which path p we choose.
    And if you’re really limber you can consider the path 2-groupoid with objects the points of X, 1-morphisms paths between points, and 2-morphisms homotopies between paths up to homotopies between homotopies. Associativity and inverses of paths now only work up to homotopy, though associativity and inverses of homotopies are strict.
    This generalizes, of course, all the way to the omega-groupoid, where omega is the first infinite ordinal. The really cool thing here is that spaces X and Y are homotopy-equivalent if and only if their fundamental omega-groupoids are equivalent.
    Yay categories.

  2. Torbjörn Larsson

    Whoa! But I think I can see some of where John is taking us. For those like me that still were mystified about going from an algebraic description of a groupoid to a categorical definition the wikipedia article cover some of that. (And I note that in turn “the category of groupoids, unlike that of groups, is cartesian closed” so they must be good for something. 🙂

  3. Norman Levitt

    If you’ll allow a critique:
    Your definitions of path and loop are a bit muddled. Given a topological space X a path in X is merely a continuous function p:[0,1] -> X . You should avoid the use of the term “smooth”, which really doesn’t make sense for topological spaces in general, but only for objects like differentiable manifolds.
    Likewise, given a base point * in X , a loop (based at *) is merely a path q:[0,1] -> X such that q(0) = q(1) = * .
    That is, the loop is a path begins and ends at * .
    One composes two loops by “chaining” them together and reparameterizing so that one gets a function on [0,1]. The operation on the loop level respects homotopy type of loops and so yields a binary operation on P(X,*), the set of homotopy classes of loops based at * (I.e, I use P for the conventional “pi-sub-one” because of typographical constraints.) The binary operation on P(X,*) makes P(X,*) into a group (not necessarily abelian).
    Asssuming X is path-connected (that is, for any x and y in X there is a path p:[0,1] -> X with p(0)=x, p(1)=y ), the isomorphism type of P(X,*) does not depend on the choice of the basepoint * , that is P(X,*_1) is isomorphic to P(X,*_2). It is important to note, however, that the isomorphism is NOT canonical. In general you show it exists by picking a path connecting base points; the isomorphisms thus obtained depend on the paths to the extent that they might be different if the two choices of path are not homotopic mod endpoints. This should have been noted.
    What makes the game go is that P is essentially a functor on the category of topolgogical spaces with base-point and base-point preserving homotopy classes of base-point preserving continuous maps (I’ll skip precise definitions here, but it essentially means that given a function f:X->Y with f(*)= *’, there is an “induced” homomorphism
    f_*:P(X,*)->P(Y,*’); moreover f_* is not sensitive to continuous deformations of f ). This also should be noted. This allows its immediate use to prove some tricky theorems, e.g., the Brouwer fixed point theorem for 2-disks and the Fundamental Theorem of Algebra.
    Going a little further afield, for X a path connected space, P(X,*) is related to the homology of X ; in particular, the integral homology group of X in dimension 1 is the abelianization of the fundamental group.
    One also notes that if X is a cell complex (e.g., a simplicial complex), then one can get an algebraic description of its fundamental group: Find a maximal tree in X (i.e., a maximal homotopy-trivial 1-dim. subcomplex). Then the fundamental group has a presentation with one generator for each 1-cell not in the tree and one relator for each 2-cell


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