Math 9120: Mapping Class Groups and Braid Groups

Fall 2016 - Spring 2017

Meets: Tue/Thu 9:30 - 10:50 in Wachman Hall, room 1036
Instructor: David Futer
Office: 1026 Wachman Hall
Office Hours: Tue/Thu 1:30 - 3:00 and by appointment
E-mail: dfuter at temple.edu
Phone: (215) 204-7854

Course content: This course will focus on the symmetry groups of manifolds and cell complexes. We will look at both the geometric and algebraic properties of the mapping class group, which can roughly be thought of as the group of symmetries of a surface. One of the ways in which we will study the group is by designing a certain cell complex on which it acts by isometries. This principle has wide generalizations: one can derive a great deal of information about a group by constructing a geometric object for the group to act on.

In the spring semester, we will focus our attention on the braid groups, which are a special class of mapping class groups. In particular, we will study the representation theory of these groups, with connections to invariants of knots and links.


Alternate sources: Prerequisites: Math 8061-62.

Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.

Detailed schedule

Day Topic Reading Homework/Note
8/30 OverviewP. 1-7
9/1 Hyperbolic geometry P. 17-23
9/6 Geodesics P. 23-26
9/8 Simple closed curves P. 26-29 Homework 1, due 9/15.
9/13 The bigon criterion P. 30-35
9/15 Arcs; Change of coordinates P. 35-41
9/20 Generating MCG by torsion elements (lecture by Dan Margalit)
9/22 MCG of the disk, punctured disk, annulus P. 46-52
9/27 MCG of the torus, punctured torus, 4-holed sphere P. 52-58
9/27 The Alexander Method P. 58-63
9/29 Dehn twist basics P. 67-72
10/4 Dehn twists and intersecton numbers P. 64-70
10/6 The center of the mapping class group P. 71-76 Homework 2, due 10/13.
10/11 Relations between Dehn twists P. 77-82
10/13 Cutting, capping, and including P. 82-88
10/18 The complex of curves P. 89-94
10/20 Complex of non-separating curves P. 94-96
10/25 Birman exact sequence P. 96-100
10/27 Finite generation P. 107-112
11/1 Lantern relation, abelianization P. 116-123
11/3 The arc complex and finite presentability P. 134-139
11/8 Symplectic basics; algebraic intersection number P. 167-173
11/10 The Euclidean algorithm P. 173-177
11/15 The symplectic representation; congruence subgroups P. 177-181, 184-187
11/17 Residual finiteness P. 187-192
11/29 Mapping class group actions on CAT(0) spaces (Thomas Ng)
12/1 Dehn surgery (Will Worden)
12/6 Mapping class group actions on character varieties (Tim Morris)
12/8 Random mapping classes (Danielle Walsh); Lickorish generators (Mark Mikida)
12/13 Moduli spaces (Elham Matinpour) in room 527

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Last modified: Fri Aug 21 13:41:22 PDT 2009