Overview: The course will be a gentle introduction to mapping class groups and their study via certain algebraic group completions. This will be done in the context of problems that arise naturally in topology, geometry and, perhaps, number theory. There will be many excursions into related topics (e.g., braid groups) to introduce background and context. Due to time constraints, some topics will only be surveyed.

The mapping class group of a compact orientable surface S is the group of connected components of the group of orientation preserving diffeomorphisms of S. Mapping class groups are important because they occur in the study of 3-manifolds (e.g., as glueing maps in Heegaard decompositions), in algebraic geometry (as fundamental groups of moduli spaces of curves) and in topological and conformal field theories. While much is known about their structure, much more has yet to be discovered.

Many non-abelian groups, such as mapping class groups and Galois groups of number fields, encode a lot of useful geometric, topological or arithmetic information which is not always easy to extract directly from the group. With most topological invariants, such as fundamental groups, there is a trade off between computability and the power of the invariant. One approach to unlocking information in a discrete group, such as a mapping class group, is to "linearize" it by replacing it by an "algebraic envelope" --- a well chosen (pro)algebraic group to which the discrete group maps. The final part of the course will be an introduction to such completions and how they can be used to give useful information about problems in geometry, topology and arithmetic.

Prerequisites: A 1-year course in algebraic topology; background in at least two of Riemannian geometry, complex analysis, representation theory and algebraic geometry is recommended.

Grading: Students will be required to do a project and perhaps make a presentation. Project topics can be chosen from related topics in topology (algebraic, geometric, symplectic), algebraic geometry, Riemann surfaces, differential geometry, number theory and other related subjects.

Worksheets:

• Worksheet 1, January 18 (pdf)
• Worksheet 2, February 11 (pdf)
• Worksheet 3, February 19 (pdf)
• Old worksheet on quaternions (pdf)

Topics:

1. Introduction and Overview

Mapping class groups; constructing elements via the classification of surfaces; Heegaard decompositions of 3-manifolds; homology manifolds; the set of Heegaard decompositions; 3-manifold invariants; surface bundles; families of algebraic curves; moduli spaces; rational points of algebraic curves.

2. Braid, Mapping Class and Torelli Groups

Configuration spaces; braid groups (classical and higher genus), generators and relations; mapping class groups; Dehn twists, complex of curves, generators and some relations, abelianization; Torelli groups, the Johnson homomorphism; Mess's theorem; survey of Johnson's results.

3. Some Homotopy Theory

Higher homotopy groups; long exact sequences of a pair and of a fibration; basic computations; cellular approximation; theorems of Whitehead and Hurewicz; Eilenberg-MacLane spaces; relation to group (co)homology.

4. Classifying Spaces

The idea of a classifying space; principal G-bundles and their classifying spaces; the case of discrete G; examples; characteristic classes.

5. Introduction to Teichmuller Theory

The uniformization theorem; hyperbolic and complex structures; pants decompositions; moduli of hyperbolic pants; marked surfaces; Teichmuller space.

6. Moduli spaces

Moduli problems; coarse and fine moduli spaces; moduli of n-pointed curves of genus 0; relation to braid groups; construction of M_g, orbifolds; moduli of elliptic curves; the classifying spaces of Diff+S and Gamama_g; character; moduli spaces of curves as algebraic varieties; their Deligne-Mumford compactifications.

7. Spectral Sequences and more Group Cohomology

spectral sequence of a filtered complex; the Leray-Serre and Hochschild-Serre spectral sequences.

8. Cohomology of Moduli Spaces

Why do we care? The Hodge bundle; tautological classes; connectivity of the complex of curves and Harer stability; applications, such as Morita-Kotschick Theorem.

9. Affine Algebraic Groups

Introduction to affine group schemes, algebraic envelopes of discrete groups.

10. Relative Completions of Mapping Class Groups

Definition, right exactness of relative completions, representation of the unipotent fundamental group of a curve; cohomological properties.

11. Arithmetic Mapping Class Groups

Profinite groups; brief introduction to arithmetic fundamental groups; the standard exact sequence.

14. Weighted Completion of Arithmetic MCGs

Weighted completion of profinite groups; weight filtrations.

12. Infinitesimal Presentations of Torelli Groups

Computation of the relative completion of mapping class groups. Applications.

References:

Topology:

1. Hatcher, Allen: Algebraic topology, Cambridge University Press, Cambridge, 2002.
2. Spanier, Edwin: Algebraic topology, Corrected reprint, Springer-Verlag, New York-Berlin, 1981.
3. Brown, Kenneth: Cohomology of groups, Corrected reprint of the 1982 original. Graduate Texts in Mathematics, 87. Springer-Verlag, New York, 1994.
4. Griffiths, Phillip; Morgan, John: Rational homotopy theory and differential forms, Progress in Mathematics, 16. Birkhäuser, Boston, Mass., 1981.
5. Whitehead, George: Homotopy theory, Massachusetts Institute of Technology, Cambridge, Mass.-London, 1966.

Braid and mapping class groups:

1. Birman, Joan: Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974.
2. Farb, Benson; Margalit, Dan: A Primer on Mapping Class Groups, Princeton Mathematical Series, Princeton University Press, 2011. (pdf, 503 pages)
3. Travaux de Thurston sur les surfaces, Séminaire Orsay. Reprint of Travaux de Thurston sur les surfaces, Soc. Math. France, Paris, 1979. Astérisque No. 66-67 (1991). Société Mathématique de France, Paris, 1991.
4. Hain, R: Moduli of Riemann Surfaces, Transcendental Aspects, Moduli Spaces in Algebraic Geometry, ICTP Lecture Notes 1, L. Gottsche editor, 2000, 293--353. (available here)
5. Farb, Benson (editor): Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, 74. American Mathematical Society, Providence, RI, 2006. (available here)

Riemann surfaces:

1. Forster, Otto: Lectures on Riemann surfaces, Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991.
2. Miranda, Rick: Algebraic curves and Riemann surfaces. Graduate Studies in Mathematics, 5. American Mathematical Society, Providence, RI, 1995.
3. Something hyperbolic

Algebraic and arithmetic geometry:

1. Hartshorne, Robin: Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2. Shafarevich, Igor: Basic algebraic geometry, 1. Varieties in projective space, Second edition. Translated from the 1988 Russian edition and with notes by Miles Reid. Springer-Verlag, Berlin, 1994.
3. Shafarevich, Igor: Basic algebraic geometry, 2. Schemes and complex manifolds, Second edition. Translated from the 1988 Russian edition by Miles Reid. Springer-Verlag, Berlin, 1994.
4. Harris, Joe; Morrison, Ian: Moduli of curves Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998.
5. Szamuely, Tamás: Galois groups and fundamental groups Cambridge Studies in Advanced Mathematics, 117. Cambridge University Press, Cambridge, 2009.
6. Matsumoto, Makoto: Arithmetic fundamental groups and moduli of curves, School on Algebraic Geometry (Trieste, 1999), 355\u2013383, ICTP Lect. Notes, 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000. (available here)

Representation theory and algebraic groups:

1. Fulton, William; Harris, Joe: Representation theory, A first course, Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991.
2. Borel, Armand: Linear algebraic groups, Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991

Weighted and relative completions:

1. Hain, Richard; Matsumoto, Makoto: Weighted completion of Galois groups and Galois actions on the fundamental group of P¹ - {0,1,infty}, Compositio Math. 139 (2003), 119--167. [math.AG/0006158]
2. Hain, Richard: Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), 597--651. (available here)