Math 569: Representations of surface groups

Emily Dumas

University of Illinois at Chicago
Spring 2017

General information

Instructor	Emily Dumas (emily@dumas.io)
Lectures	Tue & Thu, 3:30–4:45pm in Taft Hall 300
Make-up Lectures	Mon (Apr 10, 17, 24), 1:00pm in SEO 1227
CRN	39674
Office hours	Mon & Thu 2–3pm
Texts	F. Labourie, Lectures on Representations of Surface Groups. European Mathematical Society, 2013.   ISBN 978-3-03719-127-9 PDF  /  Purchase from AMS  /  Not available on Amazon  /  Not in UIC library

Course description

We will study the space of representations of the fundamental group of a surface (i.e. a real 2-manifold) into a Lie group G. We will investigate the local and global geometry of this space, including questions about connectedness, smoothness, singularities, and complex and symplectic structures. Following the main thread of the textbook, we will discuss a remarkable formula that gives the symplectic volume of the space of representations when G is compact.

After that, we will branch out and consider some special classes of surface group representations with nice geometric properties, such as representations in SL(2,R) coming from hyperbolic structures, representations in SL(2,C) associated to complex projective structures and to bending deformations of surfaces in hyperbolic 3-space, and representations in SL(3,R) associated to convex real projective structures. The idea of studying these examples is to see how some general phenomena play out in these concrete situations, and also to discuss the unique features of each one.

Course documents

• Syllabus
• Exercises   -   Last updated
• Presentation topic suggestions
• Textbook errata and clarifications
• Revised version of sections 4.2.1 to 4.2.3 (twisted de Rham isomorphism)
• Revised version of section 5.3.4

Student presentations

As described in the syllabus, each student taking the course for credit must prepare a 30-minute final presentation on a topic to be approved by the instructor.

All presentations will be in Taft 300 (the usual classroom), in three sessions:

• Tue Apr 25, in class (Lecture 29), one presentation
• Thu Apr 27, in class (Lecture 30), two presentations
• Fri May 5, 1:00-3:00pm (final exam period), three presentations

The schedule of presenters was announced by email to the class on April 6.

Links and resources

Here we collect citations for further reading about course material and links to relevant online materials.

• General references related to course material:
  • E. Witten, On quantum gauge theories in two dimensions. Comm. Math. Phys. Vol. 141, No. 1 (1991), 153-209.
    • Original reference for the volume formula.
  • W. Goldman, Geometric structures and varieties of representations.
    • Survey article; published in Contemporary Mathematics, Volume 74, American Mathematical Society, 1987.
  • W. Goldman, Locally homogeneous geometric manifolds. (ICM 2010 proceedings.)
• The classification of surfaces:
  • Gallier and Xu, A Guide to the Classification Theorem for Compact Surfaces. Springer, 2013. (Off-campus ebook access with UIC netid)
    • Very readable, with many references and historical remarks.
  • Munkres, Topology, 2ed. Prentice Hall, 2000.
    • Chapter 12 presents the classification
  • A. Putman's exposition of Zeeman's very short proof assuming triangulability
  • Hatcher's quick proof of triangulability using Kirby's torus trick
  • Ahlfors and Sario, Riemann Surfaces. Princeton University Press, 1960.
    • Sections 1-7 present the classification (interleaved with a lot of other background material)
• Character/representation varieties:
  • M. Mulase, Geometry of character varieties of surface groups
    • Includes a discussion of the finite group case.
  • A. Lubotzky and A. Magid, Varieties of Representations of Finitely Generated Groups
    • Focuses on algebraic-geometric aspects
• Fundamental group by open covers
  • A. Hatcher, Algebraic Topology. Cambridge University Press, 2002.
    • The nerve is defined in section 3.3, and the homotopy equivalence of a space with the nerve of a good cover is proved in section 4.G.
  • J. Cannon, Geometric Group Theory. Chapter 6 in Handbook of Geometric Topology, Elsevier, 2001.
    • A direct and elementary proof of the nerve description of π₁(X) for an open cover with U_i and U_i ∪ U_j simply connected is given in section 2, pp260-271.
  • Good covers of manifolds are usually constructed by taking an open cover by geodesically convex balls; existence of these is proved e.g. in:
    • Do Carmo, Riemannian Geometry, Birkhäuser, 1992. Section 3.4.
    • Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1, Publish or Perish, 1999. Chapter 9, problem 32.
• Vector bundles
  • J. Milnor and J. Stasheff, Characteristic Classes. Princeton University Press, 1974.
    • The first 36 pages give a very rapid but readable and geometric introduction to vector bundles.
  • J. M. Lee, Manifolds and Differential Geometry. American Mathematical Society, 2009.
    • See sections 6.1 to 6.5.
  • N. Steenrod, The topology of fibre bundles.
    • Homotopy theorem is Theorem 11.5
  • D. Husemöller, Fibre Bundles.
    • Homotopy theorem is Theorem 3.4.7
  • W. Ballmann, Lecture notes on vector bundles and connections
    • The correspondence between tensors and C^∞-linear maps is established in section 1.5.
• Connections and curvature
  • R. Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program
    • Chapter 3 has a detailed proof of Cartan's fundamental theorem of calculus (which we used to show that a flat connection is locally trivial).
• Construction of the moduli space
  • J. Lee, Introduction to Smooth Manifolds.
    • Chapter 21 has a detailed proof of the quotient manifold theorem for proper Lie group actions
• Moduli spaces of polygons in Euclidean space
  • M. Kapovich and J. Millson, On the moduli space of polygons in the Euclidean plane. J. Diff. Geom., 1995.
    • Definitions of the moduli spaces, length-angle duality, hyperbolic cone structure, relation to stable measures on S¹.
  • M. Kapovich and J. Millson, The symplectic geometry of polygons in Euclidean space. J. Diff. Geom., 1996.
    • Fixed-length moduli space of polygons in R³; link to moduli space of marked points on CP¹. Construction of the moduli space as a symplectic reduction. Bending flows are Hamiltonian.
  • S. Kojima, H. Nishi, and Y. Yamashita, Configuration spaces of points on the circle and hyperbolic Dehn fillings. Topology, 1999. (Also on the preprint arxiv.)
    • Detailed discussion of topology and hyperbolic structure for the equilateral case. Deformations of the edge length vector give a locally injective map to the character variety for n=5 and n=6.
  • S. Kojima, H. Nishi, and Y. Yamashita, Configuration Spaces of Points on the Circle and Hyperbolic Dehn Fillings, II. Geometriae Dedicata, 2002.
    • Global injectivity of the map to the character variety arising from deformations of edge lengths for n=5 and n=6.
  • T. Hinokuma and H. Shiga, Topology of the Configuration Space of Polygons as a Codimension One Submanifold of a Torus. Publ. Res. Inst. Math. Sci., 1998.
    • Morse-theoretic methods are used to study the topology of the moduli space of equilateral polygons (and more generally, polygons with (n-1) equal side lengths). In particular the homology and the fundamental group are computed.
  • F. Apéry and M. Yoshida, Pentagonal structure of the configuration space of five points in the real projective line. Kyushu J. Math., 1998.
    • A beautiful interpretation of the genus-four surface that parameterizes equilateral pentagons in R² as the great dodecahedron in R³.

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