# Determine monodromy representation from local system

Let $$X$$ be a path-connected manifold nice enough such it's universal covering space $$p:\widetilde{X} \to X$$ exists, $$k$$ a field. Then there exist a wellknown correspondence

$$\{\textit{linear}\text{ representations of }\pi_1(x,x)\} \leftrightarrow \{\text{local systems of }\textit{vector spaces}\text{ on }X\}$$

between $$k$$ linear finite dimensional representations of a fundamental group $$\pi_1(X,x)$$ and local systems of $$k$$ vector spaces.

The map in one direction is defined as follows: Take a $$k$$ linear rep $$\rho: \pi_1(X,x) \longrightarrow \operatorname{GL}(V)$$ where $$V$$ is a $$k$$ space and consider the associated $$V$$-bundle as quotient space $$\widetilde{X}\times_{\rho} V :=(\widetilde{X}\times V)/\pi_1(X,x)$$ where $$\pi_1(X,x)$$ acts on $$\widetilde{X}\times V$$ via

$$g \cdot(x,v) := (g \cdot x, \rho(g)\cdot v )$$

where $$g$$ acts at the left via monodromy on the covering space.

Obviously the projection to the first coordinate $$p:\widetilde{X}\times_{\rho} V \to X$$ has fiber $$V$$ and if we endow $$V$$ with the discrete topology we obtain a local system $$\mathcal{F}_{\rho}$$ on $$X$$ defined by sections

$$\mathcal{F}_{\rho}(U)= \{s:U \to p^{-1}(U) \ \vert p \cdot s =1_U \}$$

for open $$U \subset X$$. It's easy to check that if $$U$$ is contractible, then $$p^{-1}(U)\cong U \times V$$ and since $$V$$ has discrete topology, $$\mathcal{F}_{\rho}(U) \cong V$$, so it's a local system.

Question: Is there an explicit construction known to go in another direction? To start with an local system $$\mathcal{F}$$ with fibre $$V$$ and construct from it explicitly a representation $$\rho_F: \pi_1(X,x) \longrightarrow \operatorname{GL}(V)$$?

I know that it's rather easy to construct it abstractly: Let $$g=[\gamma] \in \pi_1(X,x)$$ be a class of a loop, then since $$[0,1]$$ is contractible, all local systems on $$[0,1]$$ are constant sheaves, therefore we have a chain of abstract isomorphisms

$$\gamma^*\mathcal{F}_0 \cong \gamma^*\mathcal{F}([0,1])\cong \gamma^*\mathcal{F}_1 =V.$$

Can this isomorphism of $$V$$ be written down in explicit terms as an element of $$\operatorname{GL}(V)$$ if we pick a basis $$e_1,\dotsc, e_n$$ of $$V \cong k^n$$?

Motivation of the question: In Geordie Williamson's An illustrated guide to perverse sheaves in example 5.11 one considers for $$X:= \mathbb{C}^*$$ and $$k:=\mathbb{C}$$ the covering map $$f:\mathbb{C}^* \to \mathbb{C}^*: z \mapsto z^m$$. Let $$\underline{k}$$ be the constant sheaf on $$\mathbb{C}^*$$ with value $$k=\mathbb{C}$$ regarded as 1D vector space.

One considers the pushforward sheaf $$f_*\underline{k}$$ which has as stalk at $$x=1$$ the functions from the $$m$$-set $$f^{-1}(x)$$ to $$k$$, which is isomorphic to $$k^m$$.
And then it is claimed that $$f_*\underline{k}$$ is a local system determined by the action of the monodromy on the $$m$$-th roots of $$1$$.

And I was wondering how to check this claim explicitly, even though this sounds plausible. To come back to the question I posed above it suffices to check that $$f_*\underline{k}$$ induces the repr $$\pi_1(\mathbb{C}^*,1) \cong \mathbb{Z} \to \operatorname{GL}_m(\mathbb{C})$$ which maps the generator $$1$$ to $$m$$-cycle mapping for a fixed ordered basis $$e_1,e_2,\dotsc, e_m$$ of $$k^m$$ the basis vector $$e_i$$ to $$e_{i+1}$$.

• Did you really mean $\operatorname{GL}_m(\mathcal C)$ at the end, and note $\operatorname{GL}_m(\mathbb C)$? \\ TeX note: \mathrm is not for "\rm in a math-mode environment", but for "math in \rm". For the latter, you want something like \textrm, although actually it's just \text. Note $\text{local systems}$ \text{local systems} vs. $\mathrm{local systems}$ \mathrm{local systems}. Similarly for \mathit, which in your case should have been \textit. I edited accordingly. Sep 23 at 23:49
• Starting with a local system, can you define the map as follows? Take your fiber $V$ over a basepoint $x \in X$, pull it back via a local isomorphism $p$ to a point $\tilde{x} \in \tilde{X}$ over $x$, apply a deck transformation $g$, and then define the representation to act on $V$ via $\rho(g) = \pi_* \circ g \circ (p_*)^{-1}$. If you have a reasonably explicit description of your covering map I think you can make this pretty explicit. Perhaps you run into a similar problem by trying to explicitly describe the pullback of your local system to a local system on $\tilde{X}$ though Sep 24 at 0:44
• The word "explicit" is ambiguous. "Algorithmic" is better, but then the input data has to be given in a suitable form. One way to define a flat bundle in a computable form is via a 1-cocycle for a sufficiently good finite open cover. The next issue is the fundamental group: one can describe a cover by its nerve. The 1-skeleton of the nerve will determine the group generators (once a maximal tree is chosen). Given all this, there is a relatively straightforward algorithm for computation of images of generators. Sep 24 at 2:27
• @MoisheKohan: Could you explain in a bit more details how to describe a cover from the nerve data - ie the simplicial model of classifying space - of the fundamental group as you suggest? Sep 24 at 18:54
• Later. But the direction is the opposite one: a cover determines its nerve, a cocycle defined via the cover defines a group homomorphism. Sep 24 at 19:08

Here is a way to fill in the details. For simplicity, write $$I = [0,1]$$ for the interval, and $$\exp \colon I \to S^1$$ for the function $$x \mapsto e^{2\pi i x}$$. So let $$\gamma = \exp \colon I \to S^1$$ be a generator of $$\pi_1(S^1,1)$$. To get the map $$\gamma_* \colon f_*\underline k \to f_*\underline k$$, we need to find a trivialisation of $$\gamma^*f_* \underline k$$. Luckily, this is not too hard. First, consider the pullback square $$\begin{array}{ccc}X & \stackrel p \to & S^1 \\ \!\!\!\!{\small q}\downarrow & & \downarrow{\small f}\!\!\! \\ I & \stackrel\gamma\to & S^1.\!\end{array}$$ Then $$X$$ is a disjoint union $$\coprod_{a \in \mathbf Z/m\mathbf Z} I$$ of $$m$$ copies of $$I$$ that each map isomorphically to $$I$$ under $$q$$, and $$p \colon X \to S^1$$ is given on the $$a$$-th copy by $$x \mapsto \exp((x+a)/m)$$. In other words, $$X$$ is obtained from the top copy of $$S^1$$ by breaking it up into $$m$$ pieces by cutting at the roots of unity.

Now we claim that $$\gamma^*f_*\underline k$$ is naturally isomorphic to $$q_*p^*\underline k$$. There is always a map $$\gamma^*f_*\underline k \to \gamma^*f_*p_*p^* \underline k = \gamma^*\gamma_*q_*p^*\underline k \to q_*p^*\underline k$$ coming from the unit $$1 \to p_*p^*$$ and counit $$\gamma^*\gamma_* \to 1$$ of the adjunctions $$p^* \dashv p_*$$ and $$\gamma^* \dashv \gamma_*$$ respectively. In this case, the map $$\gamma^*f_* \underline k \to q_*p^*\underline k$$ is an isomorphism, either by checking directly at stalks, or by invoking (an easy case of) the proper base change theorem.

Of course $$p^*\underline k$$ is just the constant sheaf $$\underline k$$ (this holds for pullback of any constant sheaf along any continuous map), and $$q_*\underline k$$ is isomorphic to $$\bigoplus_{a \in \mathbf Z/m\mathbf Z}\underline k$$, giving the required trivialisation. Under this isomorphism, the fibre at $$0 \in I$$ is $$\bigoplus_{a \in \mathbf Z/m\mathbf Z} \underline k_{\exp(a/m)}$$, whereas the fibre at $$1 \in I$$ is $$\bigoplus_{a \in \mathbf Z/m\mathbf Z} \underline k_{\exp((a+1)/m)}$$ (where we express everything in terms of the stalks of $$\underline k$$ in the upper copy of $$S^1$$). Both of these are isomorphic to $$(f_*\underline k)_1$$, but in ways that differ by a cyclic shift. $$\square$$

Summary. Don't just compute the isomorphism types of various vector spaces, but remember the isomorphisms. All the data is there, but only once you choose a trivialisation of $$\gamma^* \mathscr F$$.

Remark. There are alternative viewpoints that are also fruitful. For instance, locally trivialising $$\mathscr F$$ and comparing the trivialisations on the intersections gives a cocycle in $$H^1(X,\operatorname{GL}_r(\underline k))$$ (cohomology in the constant sheaf of non-abelian groups $$\operatorname{GL}_r(\underline k)$$). This set also classifies $$\operatorname{GL}_r(k)$$-torsors over $$X$$, which is the same thing as $$\operatorname{GL}_r(k)$$-covering spaces. These are again in correspondence with $$\operatorname{Hom}(\pi_1(X,x),\operatorname{GL}_r(k))$$. But we have gained a point of view using local trivialisations of $$\mathscr F$$ on $$X$$, which is often easier to work with than global trivialisations of $$\gamma^* \mathscr F$$ on $$I$$.

• concerning your remark on alternative viewpoint of the monodromy action associated to local system $\mathcal{F}$ in terms of a cocycle in $H^1(X,\operatorname{GL}_r(\underline k))$. do I understand it correctly that you pursue following stategy: Let $\gamma \subset X$ be a loop in "nice enough" space $X$. Sep 24 at 19:54
• Let $(U_i)_i^n \subset X$ be a family of open subsets of $X$ with following properties: (1) $\gamma \subset \bigcup_i U_i$ (2) $U_i$ and $U_i \cap U_{i+1}$ non empty & contractible, while $U_i \cap U_j$ for $\vert i-j \vert >1$ empty, except for $i=1, j=n$ (3)$\mathcal{F}$ trivializes over every $U_i$ (4) $U_i \cap \gamma$ contractible (5) $\gamma^{-1}(U_i)= [k_{i,1}, k_{i,2}] \subset I$ is an interval. Sep 24 at 19:54
• Then by construction of $U_i$ we have $[k_{i,1}, k_{i,2}] \cap [k_{i+1,1}, k_{i+1,2}]= [k_{i+1,1}, k_{i,2}]$ and $[k_{i,1}, k_{i,2}] \cap [k_{j,1}, k_{j,2}]$ for $\vert i-j \vert >1$ empty. Then $\gamma^* \mathcal{F}$ trivializes over each $[k_{i,1}, k_{i,2}]$. Fix now for each $i$ an explicit isom $f_j: k^n \cong \gamma^* \mathcal{F}([i_j,i_{j+1}])$ (I think here exactly appears the point you emphasised in your summary that we need an explicit choice). Sep 24 at 19:54
• We remember the datum $(g_i)_i^n$ comming from isoms $g_i: k^n \cong \gamma^* \mathcal{F}([k_{i,1}, k_{i,2}]) \cong \gamma^* \mathcal{F}([k_{i+1,1}, k_{i,2}]) \cong \mathcal{F}([k_{i+1,1}, k_{i+1,2}]) \cong k^n$ given as composition of $f_j$, can restriction and $f_{j+1}^{-1}$. Sep 24 at 19:55
• Then the associated monodromy rep should be given by mapping the class $[\gamma]$ to product $g_n \cdot g_{n-1} \cdot ... \cdot g_1$. By the the way $(g_i)_i^n$ would be the cocyce data of $\gamma^* \mathcal{F}$. Is the argument correct? Sep 24 at 19:55

In general, I would say that there is no way around the fact that the answer uses the fact that a sheaf on $$[0,1]$$ is constant.

Does this help? Instead of the pushforward of the constant sheaf $$k$$ of one-dimensional vector spaces, first think about the pushforward of the constant sheaf $$1$$ of one-element sets. Its stalk at $$x$$ is $$f^{-1}(x)$$. Think about the monodromy. In general, if $$f:E\to X$$ is any covering space, think about the monodromy of $$f_\ast 1$$ where $$1$$ is the constant singleton sheaf.

Here is another piece of advice: since you know what representation $$(V,\rho)$$ you're supposed to be getting, why not start with an explicit description of a universal covering space of $$X=\mathbb C^\ast$$ and try to see a bijection between $$\tilde X\times_\rho V$$ and the union of the stalks of $$f_\ast k$$?

• Below R. van Dobben de Bruyn's answer I gave in the comments a sketch how I would like to try to constuct the monodromy repr from cycle data of $\gamma^* \mathcal{F}$. But if I'm not missing something the same construction should go through with $\mathcal{F}:=f_*1$ since the only difference would be that in your example $\mathcal{F}:=f_*1$ would be a set valued local system, so the $g_i$ would be live in $\text{Aut}(f^{-1}(x))$, so the product $g_n \cdot g_{n-1} \cdot ...\cdot g_1$ still makes sense. Sep 24 at 19:02
• Is this the kind of approach you suggested to apply to $f_*1$ to extract the monodromy in second paragraph of your answer or did you had there another approach in mind? Sep 24 at 19:03