As a topological group, an abelian variety is just a torus. Every continuous basepoint-preserving map between tori is homotopic to a homomorphism. But the rigidity of algebraic geometry takes us further, letting us replace ‘homotopic’ by ‘equal’.
Remember that one time when I got really excited about different kinds of math coming together? (Oh wait...there were several times.) Well, get ready for a repeat. I'm excited! and it's because different kinds of math are coming together!
What happens when you cross a group with a topological space? You get a set, G, endowed with both a group structure and a topology such that:
f: G×G → G with f(x, y) = x*y
g: G → G with f(x) = x-1
are both continuous!
har har har!
Wait, that wasn't a joke? Nope. It was an actual legit definition. That is the definition of a topological group. (I actually covered the introduction to topological groups previously in a post with a LaTeX pdf, but maybe you didn't feel compelled to open the pdf. Well...you missed out! In that pdf, I also showed an interesting result about how the group and topology structures interact when they are both shoved onto a set like so; in particular, that the connected component containing the identity, e, is a normal subgroup of G.)
Continuing this topic, I'm going to show two more interesting results about topological groups.
The first is that any open subgroup of G is also closed. Yes, folks, you heard it here first: clopenness makes a comeback! Suppose H ≤ G is open. I'm going to show that G - H is also open, so then H is closed. Consider any g ∈ G - H and its coset, gH. We can define a bijection Sg: gH → H where Sg(x) = g-1*x. This function is continuous since it's just left multiplication by a fixed element, g. Since it is a bijection, g-1(H) = gH. H was open, so gH is open. Then, gH is an open neighbourhood of g; so we have G - H is open. As a result, H is closed. In fact, we know that the cosets of any subgroup partitions G into cells. Then, if G has a nontrivial proper open subgroup, G is separable and so cannot be connected.
The second cool result about topological groups is that they are homogeneous topological spaces. Definition break (much like a dance break, but for definitions):
If X is a topological space, then x and y in X are said to be swappable if there is a homeomorphism f: X → X with f(x) = y. If every pair of points in X is swappable, then X is a homogeneous space--in other words, the points in X are somehow indistinguishable from each other; open neighbourhoods around each point look exactly the same as open neighbourhoods around any other point. For example, the reals R is a homogeneous space. End of definition break!
It's pretty easy to show that a topological group G is homogeneous. Pick x and y in G, and define the new function T: G → G with T(g) = y*x-1*g. In particular, T(x) = y. That T is a homeomorphism comes out pretty easily too. T is just left multiplication by the element y*x-1, so it is continuous and has an inverse function given by T-1(g) = x*y-1*g. This inverse, of course, is left multiplication by the element x*y-1, so it is also continuous. Therefore, T is a bicontinuous bijection, so it is a homeomorphism.
We have one more class on topological groups, and I hope our postdoc continues to say cool stuff. I am so ready to get into algebraic topology and put point-set behind me...enough with compactness already!
So... I'll fill in the quotient topology stuff later. We'll do (path) connectedness and get a taste of algebraic topology today with topological groups! This stuff blew my mind because I love it when different areas of math come together (or intersect, if you will... harharhar). We didn't even cover this material in class yet; it just popped up sneakily on our homework. Our postdoc is teasing us! Can't wait to get into actual algebraic topology.
Let's talk about connected spaces! It's pretty intuitive and the formal definition isn't that difficult either. So let's jump right in. A topological space, X, is separable if you can express X = U ∪ V, where U and V are disjoint open sets. In other words, X is made up of two separate parts. If there is no separation of X, then X is a connected space.
The reason why anyone would care about connectedness is that it is a property preserved under continuous maps: in other words, it's a structural property in the category of topological groups that is preserved under the category morphism, continuous maps.
One more definition hit before I dump a pile of steaming LaTeX on you all: the connected components of a space, X, are exactly what they sound like. They are the largest subsets of X that are connected. Formally, we define an equivalence relation on X, ~, with a ~ b if there is a connected subspace of X, U, containing both a and b. The equivalence classes of this relation are the connected components. Obviously, the components of a space are disjoint connected subsets of X with the property that any connected subset of X must lie entirely in exactly one connected component.
Here's my actual writeup for the homework problem (spoiler, it has to do with ALGEBRAIC TOPOLOGY AND TOPOLOGICAL GROUPS).
There is a second related notion of connectedness, and it is path-connectedness. It's exactly what it sounds like: a space is path connected if you can draw a continuous path from one point to another, for any pair of points in the space. Formally, suppose x and y are in a topological space, X. A path from x to y is a map γ: [0, 1] → X with γ(0) = x and γ(1) = y. It's basically a parametric function whose endpoints are x and y. Of course, the interval doesn't have to be [0, 1]; it could be any closed interval [a, b] in the reals...but why make life more complicated when we could just use the easiest interval?
Similarly, the path-connected components (mouthful; on my homework, I just wrote "p-con com") of a space is a partition of the space into the largest path-connected subspaces. The formal definition basically follows the exact same format as the one for normal connectedness; just insert "path" wherever appropriate (or wherever you feel like it, really).
It turns out that path-connectedness is a stronger condition than connectedness, although for most standard spaces, they are equivalent conditions. For a counterexample of a space being connected but not path-connected, there's the (in)famous topologist's sine curve. I'm not in the business of finding esoteric counterexamples, so I'll just let Wikipedia do the work for me. Incidentally, did you know there is a book called Counterexamples in Topology? I've taken topology long enough to know that there are some pretty wacky and contrived counterexamples, but I think the fact that this book exists really just nails the point in. As a side note, my postdoc's car was broken into last weekend, and someone stole her copy of this book. Them math thieves, man.
Anyway's, here's the last bit on connectedness before I shut up and possibly write up something about the quotient topology. Or combinatorics. Or do my homework. We'd like to have a condition under which a connected space is path-connected. It may be kind of obvious/intuitive/dumb, but a space which is connected and locally path-connected is overall path-connected. Ahh...so what does it mean to be locally path connected?
A space, X, is locally-path connected if: for any point in the space, x, and any open neighbourhood of this point, Ux, there is a path connected neighbourhood Vx with the property x ∈ Vx ⊂ Ux. In other words, all open sets contain little mini path-connected open sets within them.
That's enough with me. Excuse me while I prepare the next batch of LaTeX servings.
