I guess I have no reason to offer explanation for lack of posting, but in general this has been one of the best weeks ever and at the same time one of the worst. The worst because I’ve been fairly ill and can’t seem to fully conquer it. It has been the best week for reasons I won’t mention, since I try to keep personal stuff out of this blog as much as possible (but if you know of my other blog which is purely my personal stuff, then you can read about it to your heart’s content, but I refuse to give any hints at all as to how to find that). Both of these factors has lead to a fairly unproductive week.
I may take a more algebraic topology approach for awhile. This is mainly since I’m doing a reading course on Hatcher (with two other students), and before I go present stuff to them and the prof I want to clarify my ideas.
Tomorrow I’m presenting the proof that for path connected spaces. This is a pretty wonderful result if you think about it. We have exactly how first homology and the fundamental group relate. In fact, the first thing you’d think to do (granted, this might take a little while) is the thing that works.
We can naturally think of paths and singular 1-simplices as the same thing, since they are both just continuous maps to the space out of a closed interval. So after rescaling, a loop is actually also a 1-cycle since .
The overall idea of this proof is then to show that is a well-defined homomorphism with image all of and kernel the commutator subgroup. Almost all of these facts are fairly straightforward.
First, we’ll need a few ways in which our different modes of thinking about loops versus 1-cycles correlate. If as a path , a constant, then the cycle is homologous to 0. If two paths are homotopic (in the path homotopic and hence equivalence class of sense), denoted then they are homologous. Concatenation of paths (and hence the operation in the fundamental group) is homologous to addition of the cycles (the operation in first homology). Lastly, traversing a path backwards is homologous to negating the cycle: .
So we’ll use these four facts without proof, since they are fairly standard and the proof is long enough as it is.
Recall the definition . The second fact, gives us that is well-defined since any other representative of the equivalence class will be homotopic to the original, and hence the outputs will be homologous.
The third fact gives us that is a homomorphism of groups.
Our first bit of effort comes from showing that is surjective. Here we will use the path-connected hypothesis (everything else so far is true without it). Let be any 1-cycle. We must construct a loop that maps to it.
Since the are integers, we can assume each is by just repeating the as many times as needed. But all the with -1 in front can be replaced by by the fourth property. This converts all the to 1. Thus .
But , so all the endpoints must cancel. So for any that is not a loop, in order to cancel , there must be a such that . i.e. there is some that we can concatenate with to form . In order to cancel the some other must exists with endpoint .
So we can concatenate, then rescale, and group all of these cycles into a collection of loops by the third property. So the only remaining thing we must do is get it to be a single loop. But is path-connected, so pick some basepoint . For any of these possibly disjoint loops floating around, we can pick a basepoint at each and connect with a path from to the basepoint of the i-th loop. By the third and fourth properties . So now all loops start and end at and we can combine into a single loop . Thus .
Now comes the hard part. We want . The one containment is easy. Since is abelian, by the universal property of the commutator subgroup, . The method to get the other direction is to show that for any , we must have that is trivial in the abelianization.
Suppose such that . Since is a cycle, there is some 2-chain such that . So as before, we can assume each . Now the goal is to associate a 2-dimensional -complex to by taking for each a and identifying pairs of edges which we’ll call .
So before writing this process down, we should examine what the process will be geometrically. It turns out that will be an orientable compact surface with boundary, since we are just fitting together a finite collection of disjoint 2-simplices (this is not meant to be obvious). The component containing the boundary is a closed orientable surface with an open disk removed. Since connected sums of tori can be expressed as a 2n-gon with pairs of edges identified in the manner etc, we see that is homotopic to a product of commutators.
Writing this in detail algebraically is much trickier. Given any , we have , where are singular 1-simplices. Thus .
Keep the picture of a triangle in your head. When we fit together the triangles we are getting pairs of edges. The signs on these pairs are opposite and so will cancel when we sum. The remaining (of the three sides) is a copy of . This forms our -complex .
Now form by fitting together the maps. Deform relative the edges that correspond to by mapping each vertex to . So we have a homotopy on the union of the 0-skeleton with edge , so by the homotopy extension property we get a homotopy on all of .
Now restrict to the simplices to get a new chain with boundary and loops at .
Now we just need to check whether the class is trivial or not: where . But gives a nullhomotopy of and we are done.
Thus and by the First Iso Theorem we have .