Math 9120: Mapping Class Groups and Braid Groups

Fall 2011 - Spring 2012

Meets: Tue/Thu 11:00 AM - 12:20 PM in Wachman Hall, room 527
Instructor: David Futer
Office: 430 Wachman Hall
Office Hours: 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.

Textbooks:

Prerequisites: Math 8061-62.

Class format: This will be a reading seminar, with rotating presentations by all the students.

Grading: Grades will be based on the quality of presentations, as well as participation during others' presentations.

Presentation schedule (Fall)

All page numbers refer to Farb-Margalit version 5.0, available online.

Day Topic Reading Presenter
8/30 OverviewChapter 0 Dave
9/1 Hyperbolic geometry P. 17-23 Christian
9/6 Geodesics P. 23-26 Christian, Beca
9/8 Simple closed curves P. 27-31 Beca
9/13 The bigon criterion P. 32-38 Beca, Jessie
9/15 Change of coordinates P. 38-45 Jessie
9/20 MCG of the disk, punctured disk, annulus P. 46-53 Austin
9/22 MCG of the torus, punctured torus, 4-holed sphere P. 54-59 Austin, Brian
9/27 The Alexander Method P. 59-66 Brian
9/29 Dehn twist basics P. 67-72 Christian
10/4 Dehn twists and intersecton numbers P. 72-77 Christian, Beca
10/6 The center of the mapping class group P. 77-81 Beca
10/11 Relations between Dehn twists P. 81-88 Beca, Jessie
10/13 Cutting, capping, including P. 88-92 Jessie
10/18 The complex of curves P. 94-100 Austin
10/20 Birman exact sequence P. 99-106 Austin
10/25 Finite generation P. 107-112 Brian
10/27 Explicit generators P. 112-120 Brian, Christian
11/1 Lantern relation, abelianization P. 121-128 Christian
11/3 The arc complex is contractible P. 139-141 Beca
11/8 Finite presentability P. 141-145 Beca
11/10 Symplectic basics; algebraic intersection number P. 167-173 Brian
11/15 The Euclidean algorithm P. 173-177 Brian, Jessie
11/17 The symplectic representation; congruence subgroups P. 177-181, 184-187 Jessie
11/22 Residual finiteness P. 187-192 Jessie, Austin
11/29 The Torelli group P. 193-199; Putman paper Austin
12/1 Braid groups P. 251-258 Christian
12/6 Algebraic structure of braid groups P. 258-264 Christian

Presentation schedule (Spring)

Pages are from Kassel-Turaev unless noted otherwise. FM = Farb-Margalit, CB = Casson-Bleiler.

Day Topic Reading Presenter
1/17 Braids, knots, and links P. 47-50, 58-60 Dave
1/19 An algorithm for braiding P. 61-66 Dave
1/24 Closed braids in a solid torus P. 51-55 Beca
1/26 Closed braids; Burau representation P. 56-57, 93-97 Beca, Brian
1/31 Twisted homological representations P. 98-102, 105-106 Brian
2/2 Nonfaithfulness of Burau; reduced Burau P. 102-105, 107-111 Brian, Jessie
2/7 The Alexander-Conway polynomial P. 111-118 Jessie
2/9 Lawrence-Krammer-Bigelow representation P. 118-124 Jessie, Christian
2/14 Noodles and spanning arcs P. 125-129 Christian
2/16 Faithfulness of LKB P. 137-149 Dave
2/21 Properties of the symmetric group P. 151-163 Dave
2/23 Iwahori-Hecke Algebras P. 163-169 Jessie
2/28 Ocneanu traces, HOMFLY-PT Polynomial P. 170-175 Jessie, Brian
3/1 Semi-simple algebras P. 176-192 Brian
3/13 Partitions and tableaux P. 195-203 Beca
3/15 The Young lattice P. 203-209 Beca, Christian
3/20 Simplicity of seminormal representations P. 210-217 Christian
3/22 Simplicity of reduced Burau; intro to Nielsen-Thurston theory P. 219-221; FM p. 387-390 Jessie, Brian
3/27 Measured foliations and pseudo-Anosov maps FM P. 314-321, 388-390 Brian, Dave
3/29 The Nielsen-Thurston trichotomy FM P. 390-398, 420-422 Dave
4/3 Geodesic laminations CB P. 60-68, 79-80 Dave
4/5 Stable & unstable laminations CB P. 82-86 Dave
4/10 Braids and contact geometry Beca
4/12 Braid groups in cryptography Christian
4/17 Braids, operads, and the Grothendieck-Teichmuller group Brian
4/19 Stable & unstable foliations CB, P. 89-94 Dave
4/24 Transverse measures CB, P. 95-102 Dave
4/26 The word and conjgacy problems Jessie

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