Math 801: Braids

Course description

All lectures in a single file (djvu format)

Lecture 1: Introduction. [djvu]

Lecture 2: Definitions of Braids. [djvu]

Lecture 3: Artin Braids Groups. [djvu]

Lecture 4: Fundamental Groups. [djvu]

Lecture 5: Configuration Spaces. [djvu]

Lecture 6: The Presentation Theorem. [djvu]

Lectures 7–8: The Presentation Theorem II: The Pure Braid Group. [djvu]

Lecture 9: The Dirac String Trick. [djvu]

Lectures 10–12: Mapping Class Groups. [djvu]

Lecture 12*: Mapping Class Groups of General Surfaces. [incomplete] [djvu]

Lectures 13–14: The Mapping Class Group of the Torus. [djvu]

Lectures 15–16: The Thurston–Nielsen Classification. [djvu]

Lectures 17–18: Topological Stirring. [djvu]

Lecture 19: Singularities of Foliations. [djvu]

Lecture 20: Representations of B_n. [djvu]

Lecture 21: Burau and Homology. [djvu]

Lecture 22: Topological Entropy. [djvu]

Lectures 23–24: Entropy and the Fundamental Group. [djvu]

Lecture 25: Action on π₁(M) for the Torus; Manning's Theorem. [djvu]

Lecture 26: Subshifts of Finite Type. [djvu]

Lectures 27–29: Entropy of pseudo-Anosov Diffeomorphisms. [djvu]

Lecture 30: Markov Partition for pseudo-Anosovs. [djvu]

Lecture 31: From Markov Partitions to Train Tracks. [djvu]

Lecture 32: Train Track Graphs. [djvu]

Lecture 33: Normal Train Tracks and Folding. [djvu]

Lecture 34: Measured Train Tracks and Fibered Neighbouroods. [djvu]

Lecture 35: Train Track Automata. [djvu]

Lecture 36: Train Track Automata, part II: D₄ and Culs-de-sac.

Lecture 37: Minimising the Dilatation.

Lecture 38: Maximising the Dilatation.

Lecture 39: Computer Implementation of Train Track Automata.

Bibliography and Resources

• J. S. Birman, Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.
• V. L. Hansen, Braids and Coverings, London Mathematical Society Student Texts 18, Cambridge University Press, 1989.
• Denis Auroux's lecture notes.
• D. Rolfsen, "New developments in the theory of Artin's braid groups," Topology and its Applications 127, 2003.
• J. S. Birman and T. E. Brendle, "Braids: A Survey," 2004.
• M. Epple, "Orbits of asteroids, a braid, and the first link invariant," Mathematical Intelligencer 20, 45, 1998.
• E. Artin, "Theory of braids," Annals of Mathematics 48, 101, 1947.
• B. Farb and D. Margalit, A Primer on Mapping Class Groups, version 2.95, August 2007.
• Lee Mosher's web site has several long works on mapping class groups. See also his Notices article "What is a train track?".
• W. P. Thurston, The Geometry and Topology of Three-Manifolds, Electronic version 1.1 — March 2002.
• J. Milnor, Foliations and Foliated Vector Bundles, MIT lecture notes, 1969.
• P. L. Boyland, "Isotopy Stability of Dynamics on Surfaces," 1999.
• P. L. Boyland and J. Franks, Notes on Dynamics of Surface Homeomorphisms, University of Warwick, 1989.
• G. Band and P. L. Boyland, "The Burau estimate for the entropy of a braid," 2006.
• A. Fathi, F. Laundenbach, and V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque 66–67, 1979.
• A. J. Casson and S. A. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press, 1988.
• J.-Y. Ham and W. T. Song, "The Minimum Dilatation of Pseudo-Anosov 5-Braids," Experiment. Math. 16, 167–180, 2007.
• R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, Annals of Mathematical Studies 125, Princeton University Press, 1992.

Thurston's famous cartoon of a train track.

Last modified: 13:21 2 September 2008

Background: Coarsening of a binary fluid (Simulations by Lennon Ó Náraigh)