The Standards


I’ve decided that these last couple of days I’ll post “standards” or as one professor used to say, “old standbys”. These are quick theorems that are in some sense standard in the literature, and so have at least some positive probability of showing up on a prelim exam.

Old standby 1: The fundamental group of every connected topological group is abelian. (Already a good thing I’m doing this one. I wrote “Lie group” thinking it was only true in this case, but ended up never using smooth structure).

Lemma: Let X be a topological space. Let F:I\times I\to X be continuous, and define the following paths in X: f(s)=F(s,0), g(s)=F(1,s), h(s)=F(0,s), and k(s)=F(s,1). Then f\cdot g ~ h\cdot k.

This is just annoying to actually write for how little substance this actually has. But note that f\cdot g is a path starting at F(0,0) and ending at F(1,1), as is h\cdot k. Thus it is possible to be path homotopic. Now the homotopy itself is just deforming the arguments of F through I\times I, and since F is defined and continuous through all that, it is just composing with a continuous function and hence is itself continuous.

Now for the actual problem. Fix g\in G, then \pi_1(G, g)\cong \pi_1(G, e) by the standard trick of left multiplication being a homeomorphism, so WLOG we figure out whether or not \pi_1(G, e) is abelian.

Let f,g\in \pi_1(G, e). Define F:I\times I\to G by F(s,t)=f(s)g(t) (note that this is actual group multiplication whereas in the Lemma the \cdot meant path concatenation). Since we are in a topological group, F is continuous since it is multiplication. Now by the lemma, F(s,0)\cdot F(1,s) ~ F(0,s)\cdot F(s,1). Note where f, g start and end and we get f(s)g(s) ~ g(s)f(s). Thus \pi_1(G, g) is abelian.

This set of posts might not be as useful as I thought it would be considering I left out all the parts I didn’t want to fill in, and the point is to sort of force me to go through it before actually taking the test…

Next I think I’ll do \mathbb{R}P^n is orientable if and only if n is odd.

P.S. Ack. WordPress weirdness strikes again! Who knows a hack to make a ~ without leaving the latex environment?

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6 thoughts on “The Standards

  1. Hey, I am waiting for your post about projective plane. By the way, I am just a beginner in AT, could you please show me how to calculate the $\pi_{1}(K)$ where K is the Klein bottle . Thanks and will be back

  2. There are several ways of doing it. One way uses the CW-complex representation. Think of the Klein bottle as I\times I with identifying two edges in the same way and two edges in opposite directions. This is the standard polygonal presentation. But now you can identify all the vertices to a point and get a nice presentation \langle a, b : abab^{-1}=1 \rangle.

    Another way is probably to use the covering \pi: \mathbb{R}^2\to K.

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