Braids (Spring 2008)

All lectures in
a **single
file**
[pdf]
[djvu]
(djvu format is much smaller)

Lecture 1: Introduction.

Lecture 2: Definitions of Braids.

Lecture 3: Artin Braids Groups.

Lecture 4: Fundamental Groups.

Lecture 5: Configuration Spaces.

Lecture 6: The Presentation Theorem.

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

Lecture 9: The Dirac String Trick. [

Lectures 10–12: Mapping Class Groups.

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

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

Lectures 15–16: The Thurston–Nielsen Classification.

Lectures 17–18: Topological Stirring.

Lecture 19: Singularities of Foliations.

Lecture 20:
Representations of *B _{n}*.

Lecture 21: Burau and Homology.

Lecture 22: Topological Entropy.

Lectures 23–24: Entropy and the Fundamental Group.

Lecture 25:
Action on π_{1}(*M*) for the Torus; Manning's Theorem.

Lecture 26: Subshifts of Finite Type.

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

Lecture 30: Markov Partition for pseudo-Anosovs.

Lecture 31: From Markov Partitions to Train Tracks.

Lecture 32: Train Track Graphs.

Lecture 33: Normal Train Tracks and Folding.

Lecture 34: Measured Train Tracks and Fibered Neighbouroods.

Lecture 35: Train Track Automata.

Lecture 36:
Train Track Automata, part II: *D _{4}* and Culs-de-sac.

Lecture 37: Minimising the Dilatation.

Lecture 38: Maximising the Dilatation.

Lecture 39: Computer Implementation of Train Track Automata.

- 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.