
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 
Email:  dfuter
at
temple.edu 
Phone:  (215)
2047854 
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.
Textbook:
Alternate sources:
Prerequisites: Math 806162.
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  Overview  P. 17 
 9/1  Hyperbolic geometry  P. 1723 
 9/6  Geodesics  P. 2326 
 9/8  Simple closed curves 
P. 2629  Homework 1, due 9/15.
 9/13  The bigon criterion  P. 3035 
 9/15  Arcs; Change of coordinates  P. 3541 
 9/20  Generating MCG by torsion elements (lecture by Dan
Margalit)  
 9/22  MCG of the disk, punctured disk, annulus  P. 4652 
 9/27  MCG of the torus, punctured torus, 4holed sphere 
P. 5258 
 9/27  The Alexander Method  P. 5863 
 9/29  Dehn twist basics  P. 6772 
 10/4  Dehn twists and intersecton numbers  P. 6470 
 10/6  The center of the mapping class group 
P. 7176  Homework 2, due 10/13.
 10/11  Relations between Dehn twists  P. 7782 
 10/13  Cutting, capping, and including  P. 8288 
 10/18  The complex of curves  P. 8994 
 10/20  Complex of nonseparating curves  P. 9496 
 10/25  Birman exact sequence  P. 96100 
 10/27  Finite generation  P. 107112 
 11/1  Lantern relation, abelianization  P. 116123 
 11/3  The arc complex and finite presentability  P. 134139 
 11/8  Symplectic basics; algebraic intersection number 
P. 167173 
 11/10  The Euclidean algorithm  P. 173177 
 11/15  The symplectic representation; congruence subgroups 
P. 177181, 184187 
 11/17  Residual finiteness  P. 187192 
 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
