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.](http://goodmath.blogspot.com/2006/06/group-theory-index.html). There are also some excellent articles on group theory [at Wolfram’s mathworld](http://mathworld.wolfram.com/GroupTheory.html),
and [wikipedia](http://en.wikipedia.org/wiki/Group_theory). 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:
1. ∀ a,b ∈ *O*: a*b ∈ *O* *(* is closed in O)*
2. ∀ a, b, c ∈ *O*: a*(b*c) = (a*b)*c *(* is associative)*
3. ∃ 1O, ∀ a ∈ *O*, 1O*a = a, a*1O = a. *(* has an identity value in O)*
4. ∀ 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.
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.