I'm going to start moving the topology posts in the direction of algebraic topology, which is the part of topology that I'm most interested in. There's lots more that can be said about homology, homotopy, manifolds, etc., and I may come back to it as some point, but for now, I feel like moving on.
There's some fun stuff in algebraic topology which comes from the intersection between group theory and topology. To be able to talk about that, you need the concept of a topological group.
First, I'll run through a very quick review of groups. I wrote a series of posts on group theory for GM/BM when it was at blogger; if you're interested in details, you might want to pop over there, and take a skim.. There are also some excellent articles on group theory at Wolfram's mathworld, and wikipedia. Then I'll show you the beginnings of how group theory, abstract algebra, and topology can intersect.
A Group Theory Refresher
Group theory is a branch of abstract algebra that focuses on the study of symmetry. In the group theoretical sense, symmetry is a kind of immunity to transformation. For example, if you have a square in a plane, and you mirror it around its center line, the result is indistinguishable from the square you started with: that's reflective symmetry. If you have a grid of lines, like graph paper, and you move everything one unit to the left, the result is indistinguishable from what you started with; that's translational symmetry. There are many different kinds of symmetry, but ultimately, they can all be captured using group theory in terms of an abstract multiplication operation.
A group consists of two things: a set of objects O, and a fundamental operation * : O × O → O. The group and the operator must meet the following requirements:
- ∀ a,b ∈ O: a*b ∈ O (* is closed in O)
- ∀ a, b, c ∈ O: a*(b*c) = (a*b)*c (* is associative)
- ∃ 1O, ∀ a ∈ O, 1O*a = a, a*1O = a. (* has an identity value in O)
- ∀ a ∈ O, ∃ a-1 ∈ O : a*a-1 = 1O. (Every value in O has an inverse wrt *)
Despite the fact that we generally think of the group operator as multiplication, it is not required that the group operator be commutative. If the group operator is commutative, then the group is abelian.
For a couple of quick examples of groups:
- the set of integers with the addition operator form an abelian group. 0 is the identity element; the inverse of an object x is -x.
- the set of real numbers with multiplication is not a group: 0 has no inverse. If you take the set of reals without zero, then it's an abelian group, with multiplication as the group operator, 1 as the identity value, and 1/x as the inverse of x.
- quaternions with quaternion multiplicatio form a non-abelian group.
Topological Groups
Given a group (G,×), if there is a Hausdorff topology T over the set of objects G, and × is continuous over T, then (T, ×) is a topological group.
Looking at it from the opposite point of view, if we have a Hausdorff topological space (T,τ), and we have an operation "*" over T that meets the requirements of a group operator, and:
- the operator * : T × T → T is continuous in (T,τ);
- the inverse associated with *: ()-1 : T → T is continuous in (T, τ).
then ((T, τ), ×) is a topological group. (Some people don't require the topology for a topological group to be Hausdorff; I was taught that it should be, so that's the way that I'll talk about it.)
The idea of a topological group is quite important in a lot of ways. The added structure of the group operator allows us to define what symmetry means in topological spaces; it provides a stronger form of homeomorphism and isomorphism that include symmetries. It's also the basis of a lot of more advanced structures that provide invaluable tools for analysis, and allows us to define the properties of topologies that make them useful for things like denotational semantics of programming languages.
For one example: If the topology (T, τ) in a topological group ((T, τ), ×) is a manifold, then the topological group is a Lie group. A Lie group is a basically a topological space where there is a suitable metric for doing differential calculus. Lie groups are incredibly important in analysis, and in the basic math of relativity.
Comments
Interesting exercise for the readers: show that the fundamental group of a topological group must be Abelian.
When you really grok this example you're all set for higher category theory :D.
Posted by: John Armstrong | November 13, 2006 9:01 AM
Looks like you left a semicolon off of the "×" in the pen-penultimate paragraph.
Posted by: Blake Stacey | November 13, 2006 9:55 AM
Interesting exercise for the readers: show that the fundamental group of a topological group must be Abelian.
That's quite easy. First, basic algebraic fact: when two group operations * and @ with common unit are defined on the same set and satisfy (x*y)@(u*v) = (x@u)*(y@v) they coincide and are abelian (trivial exercise). Then, the fundamental group of a topological group has an additional operation on point-wise path multiplication such that identity above holds. QED.
Posted by: nikita | November 13, 2006 10:35 AM
Okay, Nikita, now why is a 3-category with a single object and a single 1-morphism the same thing as a braided monoidal category?
Incidentally, the "trivial exercise" you mention is basically what I meant for people to solve. Yes, you know the answer, but if you feel you can freely invoke that fact then you're not the audience our host is seeking to educate.
Posted by: John Armstrong | November 13, 2006 1:41 PM
@John: If you replace "topological group" with "H-space", that was a final exam problem from the intro algebraic topology course I took last year :-)
Posted by: Cat lover | November 13, 2006 5:49 PM
I would note that it's not quite enough for a topological group to have a manifold structure for it to be a Lie group. It's important that the the binary operation and inversion be not only continuous, but differentiable to boot.
Posted by: bbs | November 13, 2006 7:16 PM
bbs: you mean that it has to be a group object in the category of manifolds? ;)
Posted by: John Armstrong | November 14, 2006 5:06 PM
To John Armstrong: I am afraid, "the audience our host is seeking to educate" has at this point not a slightest idea what fundamental group is. :-)
Yes, the trick with two group operations might seem artificial, but the whole proof is still much simpler than traditional one, invoking symmetries of S^n to prove commutativity.
Posted by: nikita | November 14, 2006 6:09 PM
nikita: I actually did mean to use the "trick", but someone should prove that the trick works as well. That trick leads into the follow-up question I asked you, which leads to a really deep and interesting understanding of both algebraic topology and categorical logic.
Posted by: John Armstrong | November 15, 2006 5:21 PM
To John Armstrong.
Yes, one can see that that trick as an instance of high-dimensional category theory fact (it should be "tricategory" rather than "3-category", by the way. Or add "weak" somewhere.). But, in addition to that, it has an absolutely elementary proof: first let y = u = 1, to obtain x@v = x*v. Then let x = v = 1, to obtain y@u = u*y. That's all.
Posted by: nikita | November 16, 2006 6:57 AM