Why are local systems and representations of the fundamental group equivalent - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:46:14Z http://mathoverflow.net/feeds/question/17786 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17786/why-are-local-systems-and-representations-of-the-fundamental-group-equivalent Why are local systems and representations of the fundamental group equivalent bavajee 2010-03-11T00:41:31Z 2013-05-03T14:30:16Z <p>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. </p> <p>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. </p> <p>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 &ge; 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". </p> <p>If there is any way this question could be improved upon, please let me know. </p> <p>Feel free to retag. </p> http://mathoverflow.net/questions/17786/why-are-local-systems-and-representations-of-the-fundamental-group-equivalent/17788#17788 Answer by Paul for Why are local systems and representations of the fundamental group equivalent Paul 2010-03-11T00:46:12Z 2010-03-11T00:46:12Z <p>Chapter 5 of <a href="http://www.ams.org/bookstore?fn=20&amp;arg1=gsmseries&amp;item=GSM-35" rel="nofollow">http://www.ams.org/bookstore?fn=20&amp;arg1=gsmseries&amp;item=GSM-35</a></p> http://mathoverflow.net/questions/17786/why-are-local-systems-and-representations-of-the-fundamental-group-equivalent/17797#17797 Answer by Donu Arapura for Why are local systems and representations of the fundamental group equivalent Donu Arapura 2010-03-11T01:41:00Z 2010-03-11T13:23:13Z <p>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.</p> <p>With regard to your other comments, perhaps I should emphasize that the Narasimhan-Seshadri correspondence is between <strong>stable</strong> 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.</p> http://mathoverflow.net/questions/17786/why-are-local-systems-and-representations-of-the-fundamental-group-equivalent/17810#17810 Answer by Omar Antolín-Camarena for Why are local systems and representations of the fundamental group equivalent Omar Antolín-Camarena 2010-03-11T03:14:56Z 2010-03-11T03:14:56Z <p><a href="http://www.math.lsu.edu/~pramod/teaching/07s-7280/notes3.pdf" rel="nofollow">Pramod Achar's notes</a> (from a lecture in an course he taught on perverse sheaves) are two pages. </p> http://mathoverflow.net/questions/17786/why-are-local-systems-and-representations-of-the-fundamental-group-equivalent/17866#17866 Answer by Frank for Why are local systems and representations of the fundamental group equivalent Frank 2010-03-11T14:17:16Z 2010-03-11T14:17:16Z <p>Tamas Szamuely's new book "Galois Groups and Fundamental Groups"</p> <p><a href="http://www.ams.org/mathscinet-getitem?mr=2548205" rel="nofollow">http://www.ams.org/mathscinet-getitem?mr=2548205</a></p> <p>contains a proof of this result and is excellently written, starting from the ground up. </p> http://mathoverflow.net/questions/17786/why-are-local-systems-and-representations-of-the-fundamental-group-equivalent/18589#18589 Answer by Magnus for Why are local systems and representations of the fundamental group equivalent Magnus 2010-03-18T13:43:42Z 2010-03-18T13:43:42Z <p>This is an old correspondence by deligne. you find it in books like:</p> <p>Voisin: Hodge theory and complex algebraic geometry I Sabbah: Isomonodromic deformations and Frobenius manifolds Kobayashi: Geometry of complex vector bundles</p> <p>It is true for local systems with complex coefficients. The rough picture is this:</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/17786/why-are-local-systems-and-representations-of-the-fundamental-group-equivalent/18649#18649 Answer by userN for Why are local systems and representations of the fundamental group equivalent userN 2010-03-18T20:20:54Z 2010-03-18T20:20:54Z <p>I'd strongly recommend having a look at Atiyah &amp; Bott's paper <em>The Yang-Mills Equations over Riemann Surfaces</em>. They explain this pretty well, and embed it in a beautiful larger picture. </p> <p>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.</p>