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My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally isomorphic to a constant sheaf. Does anyone know of a self contained, detailed treatment of this suitable for my background? I've looked at the first few pages of Delignes "Équations différentielles à points singuliers réguliers" (which my advisor suggested I take a look at) but here it just says that the equivalence is "well known", giving no reference. Neither googling ("local systems representations fundamental group") (nothing usable comes up), wiki nor the nLab entry (not detailed anough and more interested in generalisation) on local systems were of much help to me. I apologise in case the equivalence should obvious once one knows about universal covering spaces/deck transformations. I haven't learned those yet. If so, please let me know.

Why I care: I am trying to read Simpsons "Higgs bundles and local systems",. Publ. Math. I. H. E. S. 75 (1992) 5–95". Simpson assumes this equivalence but gives no references.

If there is any way this question could be improved upon, please let me know.

Feel free to retag.

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    $\begingroup$ This is the standard relationship between covering spaces and the fundamental group covered in most basic algebraic topology textbooks, no? $\endgroup$ Mar 11, 2010 at 0:43
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    $\begingroup$ Judging from the comments it seems to me that the only reason why this doesn't appear to be well covered in the literature is that you're looking in the wrong places (or insisting on keywords that aren't used in most intro algtop books). In books like Hatcher's, they use the word "bundle", not "locally constant sheaf". Instead of "local system" words like "bundle of groups" are used. Moreover it looks like you prefer not to think about bundles of groups, but the induced vector bundles from the construction Arapura describes below. $\endgroup$ Mar 11, 2010 at 3:55

7 Answers 7

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I agree that the correspondence between representations of the fundamental group(oid) and locally constant sheaves is not very well documented in the basic literature. Whenever it comes up with my students, I end up having to sketch it out on the blackboard. However, my recollection is that Spanier's Algebraic Topology gives the correspondence as a set of exercises with hints. In any case, one direction is easy to describe as follows. Suppose that $X$ is a good connected space X (e.g. a manifold). Let $\tilde X\to X$ denote its universal cover. Given a representation of its fundamental $\rho:\pi_1(X)\to GL(V)$, one can form the sheaf of sections of the bundle $(\tilde X\times V)/\pi_1(X)\to X$. More explicitly, the sections of the sheaf over U can be identified with the continuous functions $f:\tilde U\to V$ satisfying $$f(\gamma x) = \rho(\gamma) f(x)$$ for $\gamma\in \pi_1(X)$. This sheaf can be checked to be locally constant. Essentially the same procedure produces a flat vector bundle, i.e. a vector bundle with locally constant transition functions. This is yet another object equivalent to a representation of the fundamental group.

With regard to your other comments, perhaps I should emphasize that the Narasimhan-Seshadri correspondence is between stable vector bundles of degree 0 and irreducible unitary representations of the fundamental group. The bundle is constructed as indicated above. Anyway, this sounds like a good Diplom thesis problem. Have fun.

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    $\begingroup$ To give an alternative formulation of the same thing: One could consider the universal cover as a principal $\pi_1(X)$-bundle on X, then there exists a cover such that the transition functions $\gamma_{ij}$ (which are elements of $\pi_1$) of this bundle generate $\pi_1$, and one then gets transition functions on the associated bundle/local system by $\theta_{ij}:=\rho(\gamma_{ij})$. (I believe this works, but correct me if I'm wrong.) $\endgroup$ Mar 12, 2010 at 15:20
  • $\begingroup$ Yes, absolutely right. $\endgroup$ Mar 13, 2010 at 0:30
  • $\begingroup$ @DonuArapura Hi, Donu, I have a naive question about conventions. Since fundamental group acts on the right, should it be $f(x \gamma )=f(x) \rho (\gamma)$? Has the math community adopted the convection to write everything on the left when talking about things like monodromies? $\endgroup$
    – Wenzhe
    Mar 1, 2017 at 17:37
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Tamas Szamuely's new book "Galois Groups and Fundamental Groups"

http://www.ams.org/mathscinet-getitem?mr=2548205

contains a proof of this result and is excellently written, starting from the ground up.

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Pramod Achar's notes (from a lecture in an course he taught on perverse sheaves) are two pages.

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This is an old correspondence by deligne. you find it in books like:

Voisin: Hodge theory and complex algebraic geometry I Sabbah: Isomonodromic deformations and Frobenius manifolds Kobayashi: Geometry of complex vector bundles

It is true for local systems with complex coefficients. The rough picture is this:

A flat connections is equivalent to a local system and the parallel transport of this connection in a loop only depends on the loop, so gives you a represenation of the fundamental group.

A representation rep of the fundamental group of X defines a complex vector bundle of rank r via X x C^{r}/~ (fibers identified by rep). This carries again a flat connection.

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Here is the way I think of the correspondence between locally constant sheaves and representations of the fundamental group, and how I like to tell my students about it when I introduce it in class (the expected background being the fundamental group, the theory of covering spaces, and the notion of sheaf).

Sheaves as sheaves of sections of étalé spaces

First, there is a correspondence

$$ \mathrm{Sh}_X:=\{\mathrm{sheaves\ of\ sets\ on}\ X\} \leftrightarrow \mathrm{Et}_X := \{\mathrm{local\ homeomorphisms}\ p:Y\longrightarrow X\} $$

sending a sheaf $\mathcal{F}$ on $X$ to the étalé space $$\mathrm{Et}(\mathcal{F}) := \bigsqcup_{x\in X} \mathcal{F}(x)\, ,$$ where $\mathcal{F}(x) := \varinjlim_{U\ni x} \mathcal{F}(U)$ is the stalk of $\mathcal{F}$ at $x$, and an étalé space $p:Y\longrightarrow X$ to the sheaf of (continuous) sections $$U\longmapsto\Gamma_Y(U):=\{s:U\longrightarrow Y\ |\ p\circ s=\mathrm{id}_U\}.$$

This is useful already in the basic theory of sheaves, for instance to construct the sheaf associated to a pre-sheaf. One possible reference is:

Sections of covering spaces

Next, under this correspondence, locally constant sheaves correspond precisely to covering spaces:

$$ \mathrm{Loc}_X:=\{\mathrm{locally\ constant\ sheaves\ of\ sets\ on}\ X\} \leftrightarrow \mathrm{Cov}_X := \{\mathrm{covering\ map}\ p:Y\longrightarrow X\} $$

Indeed, a locally constant sheaf is locally isomorphic to the sheaf of continuous sections of a product space $X\times F$, where $F$ is a discrete topological space (in particular, such sections are locally constant maps, with values in $F$).

The upshot of working with a covering map $p:Y\longrightarrow X$ is that (since $X$ is nice) we can lift paths in $X$ (and homotopies between them) to $Y$, in a unique manner. In particular, there is a well-defined map

$$\rho: \pi_1(X,x) \longrightarrow \mathrm{Aut}(Y_x)$$ (where $Y_x:=p^{-1}(\{x\})$ is the fibre of $p$ above $x$) defined by sending the homotopy class of a loop $\gamma:[0;1]\longrightarrow X$ at the base point $x\in X$ to the bijective transformation $$\rho(\gamma): (y \in F) \longmapsto \widetilde{\gamma}^{(y)}(1) \in F$$ where $\widetilde{\gamma}^{(y)}:[0;1]\longrightarrow Y$ is the unique continuous map such that $p\circ\widetilde{\gamma}^{(y)}=\gamma$ and $\widetilde{\gamma}^{(y)}(0)=y$.

The correspondence

With an appropriate convention on the composition of paths in $X$, the map $\rho$ becomes a group morphism. So, given a discrete topological space $F$, the choice of base point $x\in X$ induces a map

$$\Phi:\{\mathrm{covering\ spaces\ of}\ X\ \mathrm{with\ fibre}\ F\} \longrightarrow \mathrm{Hom}\big(\pi_1(X,x);\mathrm{Aut}(F)\big)$$

The converse map

$$\Psi: \mathrm{Hom}\big(\pi_1(X,x);\mathrm{Aut}(F)\big) \longrightarrow \{\mathrm{covering\ spaces\ of}\ X\ \mathrm{with\ fibre}\ F\}$$

is the map defined in Donu Arapura's answer: a group morphism $\rho: \pi_1(X,x) \longrightarrow \mathrm{Aut}(F)$ is sent to the covering space

$$(\widetilde{X}\times F)\,/\,\pi_1(X,x) \longrightarrow \widetilde{X}\,/\,\pi_1(X,x) = X$$

where, for $X$ nice, $\widetilde{X}$ is the universal covering space of $X$ (determined up to canonical isomorphism by the choice of the base point $x\in X$) and $\gamma\in\pi_1(X,x) \simeq \mathrm{Aut}_X(\widetilde{X})$ acts on $(\xi,v)\in(\widetilde{X}\times F)$ via $$\gamma\cdot(\xi,v) := \big(\gamma\cdot\xi, \rho(\gamma)\cdot v\big).$$

Local systems of vector spaces

Once it is checked that this is indeed a covering space of $X$ with fibre $F$, it remains to prove that the maps $\Phi$ and $\Psi$ are indeed inverse to each other.

This is compatible with the notion of isomorphisms of covers and equivalence of representations, so it provides a bijection

$$\check{H}^1\big(X;\mathrm{Aut}(F)\big) \simeq \mathrm{Hom}\big(\pi_1(X,x);\mathrm{Aut}(F)\big)\, \big/\, \mathrm{Aut}(F)$$

where $\check{H}^1\big(X;\mathrm{Aut}(F)\big)$ is the set of isomorphism classes of locally trivial bundles with (discrete) fibre $F$ and structure group $\mathrm{Aut}(F)$ over $X$.

When $F=V$ is a finite-dimensional vector space equipped with the discrete topology, you can restrict the above to the subgroup $\mathbf{GL}(V) \subset \mathrm{Aut}(V)$ and obtain a correspondence

$$\{\mathrm{local\ systems\ of}\ \mathit{vector\ spaces}\ \mathrm{on}\ X\} \leftrightarrow \{\mathit{linear}\ \mathrm{representations\ of}\ \pi_1(x,x)\}.$$

The notation $\check{H}^1\big(X;\mathbf{GL}(V)\big)$ is commonly to designate the set of isomorphism classes of flat vector bundles with fibre $V$ on $X$ (flat in the sense that the transition functions $$g_{U_2U_1}:U_1 \cap U_2 \longmapsto \mathbf{GL}(V)$$ are locally constant on the open set $U_1 \cap U_2\subset X$).


To go further, if the topological space $X$ is a real or complex manifold $(X,O_X)$, the correspondence between locally constant sheaf of vector spaces and linear representations of the fundamental group can also be phrased in terms of vector bundles equipped with an integrable connection.

Namely, as in this answer to a related question on MO, a local system of vector spaces $\mathcal{V}$ is sent to the vector bundle ($=$ locally free $O_X$-module) with integrable connection

$$(\mathcal{E}:= O_X \otimes_{\mathrm{Cst}_X} \mathcal{V}\, ,\ \nabla:= d \otimes\mathrm{id})$$

where $\mathrm{Cst}_X\subset O_X$ is the sheaf of locally constant functions on $X$ (with values in $\mathbb{R}$ or $\mathbb{C}$) and $d:O_X\longrightarrow \Omega^1_x$ sends a function $f\in O_X(U)$ to the $1$-form $df\in\Omega^1_X(U)$, so

$$\nabla:\mathcal{E}\longrightarrow \Omega^1_X \otimes_{O_X} \mathcal{E}$$

is indeed a linear connection on $\mathcal{E}$ (=it satisfies the Leibniz identity $\nabla(f\cdot s) = df\otimes s + f\cdot \nabla s$ for all open set $U\subset X$, all section $s\in\mathcal{E}(U)$ and all function $f\in O_X(U)$).

In this picture, the local system of vector spaces $\mathcal{V}$ is viewed as a locally free $\mathrm{Cst}_X$-module, which is why the operation $$\mathcal{V}\longmapsto O_X \otimes_{\mathrm{Cst}_X} \mathcal{V}$$ makes sense.

The converse map sends the vector bundle with integrable connection $(\mathcal{E},\nabla)$ to the sub sheaf $$U\longmapsto\mathcal{E}^\nabla(U) := \{s\in \mathcal{E}(U)\ |\ \nabla s=0\},$$ which is a locally constant sheaf (the sheaf of locally constant sections of $\mathcal{E}$, which is locally isomorphic to the sheaf of differentiable sections of $X\times V$, where $V$ is a finite-dimensional real or complex vector space, endowed here with its usual topology).

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  • $\begingroup$ Welcome to mathoverflow! $\endgroup$ Sep 24 at 16:21
  • $\begingroup$ Thank you David! $\endgroup$ Sep 25 at 6:33
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I'd strongly recommend having a look at Atiyah & Bott's paper The Yang-Mills Equations over Riemann Surfaces. They explain this pretty well, and embed it in a beautiful larger picture.

The basic idea is that a locally constant sheaf can be viewed as the horizontal sections of a bundle with respect to a flat connection. The holonomy of a flat connection along a curve only depends on the homotopy class of the curve, hence gives a representation of the fundamental group.

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Chapter 5 of James F. Davis and Paul Kirk, Lecture Notes in Algebraic Topology (AMS, Graduate Studies in Mathematics 35).

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