I'm trying to understand how to use the Mayer-Vietoris sequence to compute Cohomologies. There's a small chapter in Tu's Introduction to Manifolds explaining the basics, with some basic theory.
More specifically section 26.2 has an example about the circle, later on there're other examples. There's the following table which I don't quite understand how to read it, I also don't understand how it's filled:
The questions are:
- How am I supposed to read the table?
- How is it filled exactly? (I do understand the author covers $S^1$ with two overlapping circle arcs, which allows him to build a short exact sequence (I think this should correspond to a row of the table?) I think he's also using the zig-zag lemma, but not sure how. The exact entries are a bit of a mistery to me, especially all those zeros.
- Is the Mayer-Vietoris sequence supposed to simplify the computation of cohomologies? With reference to this specific example to me what appears to happen in a sense is that the author still uses some of the previous results to fill some of the entries. However by the use of problem 26.2 he's able to compute the dimension of H^1(S^1) right away, without passing through the calculations done in the example 24.4. But it doesn't seem to me it simplified that much.
Can you clarify?
Assuming you're familiar with the Mayer-Vietoris sequence, you know it goes $0 \to 1 \to 2$ in grading, so you read each row left to right, and you start from the bottom and go to the top.
Each point in the table is filled with the homology group of the degree from the row, and the space from the column. The whole point of MV is to use simpler spaces for which you know the cohomology. In this case, these are two arcs that intersect in two points. Since arcs are contractible, their cohomologies are $\mathbb{R}$ in degree $0$ and $0$ otherwise. Usually you use a short exact sequence of (co)chain complexes, specifically $$0 \to C^n (X \cup Y) \xrightarrow{i} C^n(X \coprod Y) \xrightarrow{j} C^n (X\cap Y) \to 0$$ and apply the zig-zag lemma to get Mayer-Vietoris in the first place, so you don't actually have to go through this. The actual Mayer-Vietoris sequence has all the terms in cohomology already written out, so all you have to do is plug and play.
Usually it's a pretty useful tool. Here the space is $S^1$ so it's already pretty simple. But suppose I asked you to calculate the de Rham cohomology of $\mathbb{R}^2 - \{p,q\}$ i.e. a plane punctured twice? MV could help you (though there are other ways for sure). Just checked the book and on page $302$ he calculates the cohomology groups of a torus using MV, that could be illuminating. After further skimming, a lot of chapter $28$ can help illustrate the power of MV.