I'm currently reading "Differential Forms in Algebraic Topology" by Bott & Tu. Unfortunately, I do not have much backgrounds in this field, so my question might be due to missing some very basic concepts.
In chapter 13 Monodromy, the author proves two theorems :
- If $\pi_{1}(N(\mathfrak{U}))$, the fundamental group of the nerve of a good cover, is equal to zero, then every locally constant presheaf on $\mathfrak{U}$ is constant.
- For a topological space $X$ with good cover $\mathfrak{U}$, we have $\pi_{1}(X)\simeq \pi_{1}(N(\mathfrak{U}))$.
I did have a difficult time understanding the proofs of the two theorems (not quite sure that I actually did understand them), but what really frustrates me are the examples and exercises at the end of the section:
Example 13.5
Let $S^{1}$ be the unit circle in the complex plain with good cover $\mathfrak{U}=\{U_{0}, U_{1},U_{3}\}$ as in the picture below.
The map $\pi : z \rightarrow z^2$ defines a fiber bundle $\pi : S^1 \rightarrow S^1$ each of whose fibers consists of two distinct points. Let $F=\{A,B\}$ be the fiber above the point 1.
The cohomology $H^*(F)$ consists of all functions on $\{A,B\}$, i.e., $H^*(F)=\{(a,b)\in \mathbb{R}^2\}$ . (I don't understand this part. Why is this true? Could someone explain?)
The author proceeds and mentions that the presheaf $\mathcal{H}^*(U)=H^*(\pi^{-1}U)$ is not a constant presheaf.
Exercis 13.6 tells me to compute the Čech cohomology $H^*(\mathfrak{U},\mathcal{H}^0)$ directly, noting that $H^*(S^1)=H^*_D\{C^*(\pi^{-1}\mathfrak{U},\Omega^*)\}=H_{\delta}H_d=H^*(\mathfrak{U},\mathcal{H}^0)$.
By using "indirect" methods, I would get $H^0(\mathfrak{U},\mathcal{H}^0)=H^1(\mathfrak{U},\mathcal{H}^0)=\mathbb{R}$ from above, right?
But I'm having trouble calculating the cohomology directly. In order to compute, I think I'd need to know:
- what $C^0(\mathfrak{U},\mathcal{H}^0)$ and $C^1(\mathfrak{U},\mathcal{H}^0)$ looks like.
- what the transition map $\rho$ 's looks like.
- how the difference operator $\delta_0$ behaves
- the image and kernel of $\delta_0$
From $H^*(F)=\{(a,b)\in \mathbb{R}\}$, I guess $C^0(\mathfrak{U},\mathcal{H}^0)$ and $C^1(\mathfrak{U},\mathcal{H}^0)$ would be something like $\mathbb{R}^2 \oplus \mathbb{R}^2\oplus \mathbb{R}^2$, but from then on, I'm completely lost.
In addition, I don't see how this is related to monodromy dealt in this chapter.
I'm pretty sure that these questions are quite easy to answer, but could someone give me a in-depth explanation? I would like to solve the next two problems on my own!
Best Answer
As you said, $C^0(\mathfrak{U},\mathcal{H}^0)=\mathbb{R}^2\oplus\mathbb{R}^2\oplus\mathbb{R}^2$ and the same isomorphism holds for $C^1$. Now, we need to compute the differential, and for this, we need to understand the restriction map.
Recall that $\pi:\pi^{-1}(U)\to U$ is the map $z\mapsto z^2$. In other words, $\pi^{-1}(z)$ is the set of square roots of $z$. So if $U$ is an arc, $\pi^{-1}(U)$ is the union of two arcs and you can think of it as follows : if you pick $z\in U$, you have two choices of square roots, now when $z$ moves in $U$, the two choices moves continuously and forms two arcs.
Now let $z_0\in U_0\cap U_1, z_1\in U_1\cap U_2$ and $z_2\in U_2\cap U_0$. Write $z_i^a, z_i^b$ for the two choices of square roots of $z_i$. We won't make arbitrary choices for them. Instead, we will define them as follow :
So now, you need to see the following :
Thus, using the identifications $\mathcal{H}^0(U_0)=maps(\{z_0^a,z_0^b\},\mathbb{R})$ and so on, we see that
It follows that the operator $\delta_0$ is given by the block matrix (each block is a $2\times 2$ matrix) $$ \begin{pmatrix} Id & Id & 0\\ 0& -Id & Id\\ -\tau& 0& -Id \end{pmatrix}$$