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 be a topological space. Let be continuous, and define the following paths in X: , , , and . Then ~ .
This is just annoying to actually write for how little substance this actually has. But note that is a path starting at and ending at , as is . Thus it is possible to be path homotopic. Now the homotopy itself is just deforming the arguments of through , and since 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 , then by the standard trick of left multiplication being a homeomorphism, so WLOG we figure out whether or not is abelian.
Let . Define by (note that this is actual group multiplication whereas in the Lemma the meant path concatenation). Since we are in a topological group, is continuous since it is multiplication. Now by the lemma, ~ . Note where start and end and we get ~ . Thus 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 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?