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i am trying to understand the following example:

A locally constant sheaf of $\mathbb{C}$-vector spaces of finite rank on $X$ ist called a local system! Hence it is now possible to classify all local systems on the sapce $X := \mathbb{R}^2 \setminus 0$. In fact, $\pi_1(X) \cong \mathbb{Z}$, hence, $Hom(\pi_1(X),GL(\mathbb{C^n})) = GL(\mathbb{C}^n)$. A local system $F$ of rank $n$ is determinded, up to isomorphism, by its monodromy $\mu(F) \in GL(\mathbb{C}^n)$. The classification of such sheaves is thus equivalent to that of invertible $n \times n$ matrices over $\mathbb{C}$ up to conjugation.

In particular, when $n=1, GL(\mathbb{C}^n) = \mathbb{C}^{\times}$. Hence, a local system of rank one is determined, up to isomorphism, by its monodromy $\alpha \in \mathbb{C}^\times$.

I am struggeling with a few things here:

(i) Why is $Hom(\phi_1(X),GL(\mathbb{C^n})) = GL(\mathbb{C}^n)$?? And what would it be if it wasnt isomorphic to $\mathbb{Z}$

(ii) How does an $\alpha$ would look like?

It would be great if someone could give a concrete example of what is meant here, like, as simple as it gets.

1 Answers 1

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I assume $\phi_1=\pi_1$ ?

(i) The first question is really easy : $Hom(\pi_1(X),GL(\mathbb{C}^n))=Hom(\mathbb{Z},GL(\mathbb{C}^n))=GL(\mathbb{C}^n)$ because a morphism $\mathbb{Z}\rightarrow GL(\mathbb{C}^n)$ is entirely determined by the image of $1\in\mathbb{Z}$. This image is denoted $\alpha$ and is the monodromy operator.

If $\pi_1(X)\neq\mathbb{Z}$, then of course, we don't have such isomorphism. But for example, if $X=\mathbb{T}^2$ is a torus, $\pi_1(X)=\mathbb{Z}^2$ and $Hom(\pi_1(X),GL(\mathbb{C}^n))$ is isomorphic to the set of pairs $(f_1,f_2)$ where $f_1$ and $f_2$ are two commuting linear isomorphism of $\mathbb{C}^n$. ($f_1$ and $f_2$ are the image of the canonical basis of $\mathbb{Z}^2$).

For general $X$, the data of a local system is equivalent to a representation of $\pi_1(X)$.

(ii) Let $X=\mathbb{C}^\times$. A local system of rank 1 can be pictured as follow : for each point $x\in X$, imagine that we have a copy $F_x$ of $\mathbb{C}$ (with the discrete topology). These copies of $\mathbb{C}$ are not unrelated : if $U$ is a small disc in $X$, $x,y$ are two points in $U$, then we can identify $F_x$ and $F_y$. The identification can be pictured as follow : if $c:x\rightarrow y$ is a path from $x$ to $y$ in $U$, then for each $f_x\in F_x$, there is only one way to move from $f_x$ in the local system such that "we stay over the path $c$" (this is because $F_x$ is discrete !). We end up in $F_y$ to get an element $f_y$. The correspondence $f_x\mapsto f_y$ is the identification (it doesn't depend on the choice of the path $c$).

But what happen if I take two points $x,y$ not closed to each others ? We can take a path from $x$ to $y$, and we will have again an identification between $F_x$ and $F_y$. But this time, this identification will depends on the choice of the path (at least, two homotopic paths will give the same identification).

So if I take a loop based at $x$, I will get an automorphism of $F_x$, this isomorphism may not be the identity ! This is the monodromy along the loop. It is an important fact that the monodromy along every (up to homotopy) loops characterize the local system.

On $\mathbb{C}^\times$, this is entirely determined by the monodromy around the origin, this is your $\alpha:F_x\rightarrow F_x$.

(iii) Let's see a concrete example. Let $F$ be the local system such that $F(U)$ is the set of solutions to the differential equation of $\frac{f'}{f}=\frac{1}{2z}$ on $U$. At a point $x\in X$, there is a bijection $F_x\simeq \mathbb{C}$ given by $f\mapsto f(x)$.

Note however, that if $x,y$ are two points in a small disk, the identification $F_x\simeq F_y$ is the following : let $f_x\in F_x\simeq\mathbb{C}$, let $f$ be a solution of $\frac{f'}{f}=\frac{1}{2z}$ on $U$ such that $f(x)=f_x$, then $f_y\in F_y$ is simply $f(y)$ through the identification $F_y\simeq\mathbb{C}$.

I am not sure I am very clear here (please tell me !).

Now I claim that if $c$ is a loop around the origin, the map $F_x\rightarrow F_x$ is the multiplication by $-1$. Indeed, the solutions of the differential equation on a small disk $U$ are the functions $C\sqrt{z}$ for a given choice of square root of $z$. But if we turn around the origin, we will get the opposite choice.

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    First of all, i want to thank you for that awesome response to my question. There are a few questions left which i am going to order to the topics you wrote. (i) So the fact that i have just the coise of $\mathbb{Z} \longrightarrow GL(\mathbb{C^2})$ $1 \longmapsto [1 0; 0 1]$ determines that isomorphism? – 2017-02-17
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    (ii) (1) By $F_x$ i assume you are talking about the Stalk at point $x$ (2) Why do we have a copy of $\mathbb{C}$ rather then a copy of $\mathbb{C}^\times$ if this is acutally the space? (3) Why is $F_x$ equipped with the discrete topology? (4) For some far away points, say $x = 1$ and $y = -1$, the path around the origin in positiv direction and the path around the origin in negative direction are not homotopic. So these two paths give a different identification? (5) So THE MONODROMY is just the morhism $\alpha : F_x \rightarrow F_x$ ? – 2017-02-17
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    For (i), this is not what I meant : $Hom(\mathbb{Z},GL(\mathbb{C}^2))\overset{\sim}\rightarrow GL(\mathbb{C}^2)$ via the map $(f:\mathbb{Z}\rightarrow GL(\mathbb{C}^2))\mapsto f(1)$. The inverse of this map is $M\mapsto (n\mapsto M^n)$. Check that these map are inverse to each other. – 2017-02-17
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    For (ii)(1), yes $F_x$ is the stalk at $x$. (2) We have a copy of $\mathbb{C}$ because $F$ is sheaf of $\mathbb{C}$-vector spaces, and the stalk are dimension 1 $\mathbb{C}$-vector spaces. (3) $F_x$ is just a set, the discrete topology is simply here to give a picture. If $F$ was the trivial local system, $F(X)=\mathbb{C}$, in other words, $F(X)$ will be the constant maps $X\rightarrow\mathbb{C}$ (or the continuous maps $X\rightarrow\mathbb{C}$ if we give the target the discrete topology). – 2017-02-17
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    It is important in the identification $F_x\simeq F_y$. In fact, if $U$ is a small disk, $F$ is isomorphic to the sheaf of section of $U\times\mathbb{C}\rightarrow U$ where $\mathbb{C}$ is given the discrete topology. This is why a path $x\rightarrow y$ in $U$ and a starting point $(x,f_x)$ can be lift uniquely. (4) Yes in general two non homotopic paths give two non different identifications. (5) In general, the monodromy at $x$ is the action of the whole $\pi_1(X,x)$. But if $\pi_1(X,x)\simeq\mathbb{Z}$, it is completely determined by the action of the generator 1. – 2017-02-17
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    So, coming back to the first example, this would looke like $\alpha : \mathbb{Z} \longrightarrow \mathbb{C}^\times$ $1 \mapsto \alpha(1) := exp(1* 2 \pi i)$, where $1$ is the loop around the origin in positive direction. or is it a morphism like this $\alpha : \mathbb{Z} \longrightarrow \mathbb{C}^\times$ $1 \mapsto \alpha(1) := exp(1* 2 \pi i \alpha)$ because i also found something like the last one – 2017-02-17
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    i am sorry i souldnt have used $\alpha$ in the second on. Would more look like $\rho : \mathbb{Z} \longrightarrow \mathbb{C}^\times$ $1 \mapsto \rho^{\alpha}(1) := exp(1* 2 \pi i \alpha)$ because i also found something like the last one – 2017-02-17
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/53804/discussion-between-roland-and-e-j-k). – 2017-02-17