## Why are local systems and representations of the fundamental group equivalent

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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.

My background: I am a Diplom (roughly equivalent to MSc) student at a german university. For my Diplom thesis I am aiming to understand Narasimhan and Seshadris Theorem (from "Stable and unitary vector bundles on a compact Riemann surface") which I think roughly states that a holomorphic vector bundle of degree zero on a compact Riemann surface X of genus g ≥ 2 is stable if and only if it arises from an irreducible unitary representation of the fundamental group of X. I also hope to read and understand parts of Hitchins paper "The Self-Duality Equations on a Riemann Surface" (Proc. London Math. Soc. 1987 s3-55: 59-126) and as mentioned above parts of Simpsons article "Higgs bundles and local systems".

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

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This is the standard relationship between covering spaces and the fundamental group covered in most basic algebraic topology textbooks, no? – Ryan Budney Mar 11 2010 at 0:43

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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. – Ryan Budney Mar 11 2010 at 3:55

## 6 Answers

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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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.) – Ketil Tveiten Mar 12 2010 at 15:20
Yes, absolutely right. – Donu Arapura Mar 13 2010 at 0:30

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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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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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