Math 8803

Torelli groups

Spring 2018



Dan Margalit

Class Meetings

Monday, Wednesday, and Friday 10:10-11:00 am, Skiles 368.

Office Hours

By appointment.


Students will be expected to complete one assigment per month of their choosing. Possible assignments include: exercises from lecture, written summaries of extra readings, mini-lectures in class, lectures in the student seminar. Group work is encouraged.


The material for the course will be divided into six topics, as follows. In the course we will make contact with geometric group theory, algebraic topology, algebraic geometry, representation theory, and the theories of 3-manifolds and 4-manifolds.

Basics: Generation of the mapping class group by Dehn twists, the symplectic representation of the mapping class group, separating twists and bounding pair maps, conjugacy classes, the lantern relation and its relatives, torsion freeness

Generation: the complex of cycles, generation by bounding pair maps, finite generation, cubic generation, genus two

The Johnson homomorphism: three definitions, the Chillingworth homomorphism, exponential distortion, generating the kernel, finite generation of the kernel

The abelianization: Birman-Craggs-Johnson homomorphisms and quadratic forms, the abelianization, the Casson invariant and the Johnson homomorphism, Pitsch's theorem

Higher finiteness properties: Torelli space and the Torelli theorem, Non-finiteness of cohomology in genus 2 and genus 3, Akita's non-finiteness, cohomological dimension

Representation stability: Parameterized Abel-Jacobi maps, Generating the Johnson filtration, finite generation of second homology as an Sp-module

Weekly Schedule

Week Dates Topics Reading Lectures Notes
1 Jan 8-12 Overview / MCG Primer Intro MCG 1 No class Mon ("ice storm")
2 Jan 15-19 MCG Primer MCG 2 MLK day + snow day
3 Jan 22-26 Sp rep Primer Sp rep
4 Jan 29-Feb 2 Complex of cycles Hatcher-M
Torelli gens
5 Feb 5 - Feb 9 Generating Torelli Hatcher-M
6 Feb 12 - Feb 16 Generating Torelli Mess
Johnson I
Johnson gens
7 Feb 19 - Feb 23 Johnson homomorphism Johnson Johnson hom
8 Feb 26 - Mar 2 Chillingworth class / BCJ Chillingworth 1
Chillingworth 2
Johnson Quadratic
9 Mar 5 - Mar 9 BCJ / ZHS3 Morita I Homology 3-spheres
10 Mar 12 - Mar 16 Johnson kernel Johnson II J II
10 Mar 26 - Mar 30 Abelianization Johnson III J III
11 Apr 2 - Apr 6 Abelianization Johnson III J III
12 Apr 9 - Apr 13 pseudo-Anosovs / Torelli space
13 Apr 16 - Apr 20 Kg is fg Church-Ershov-Putman Kg
14 Apr 23 Problems Problem list Problems
Course notes

Homework suggestions


---A primer on mapping class groups, Benson Farb and Dan Margalit

---Office hours with a geometric group theorist, Matt Clay and Dan Margalit

---Thurston's work on surfaces, Djun Kim and Dan Margalit

---A survey of the Torelli group, Dennis Johnson

---Lectures on the Torelli group, Andrew Putman

---Geometry, Topology, and the Torelli group, Benson Farb and Nick Salter

---Mapping class groups, Nikolai Ivanov

---Generating the Torelli group, Allen Hatcher and Dan Margalit

---The dimension of the Torelli group, Mladen Bestvina, Kai-Uwe Bux, and Dan Margalit

---The Torelli groups for genus 2 and 3 surfaces, Geoffrey Mess

---The structure of the Torelli group I: a finite set of generators for I, Dennis Johnson

---An abelian quotient for the mapping class group I, Dennis Johnson

---Winding numbers on surfaces, I., D.R.J. Chillingworth

---Winding numbers on surfaces, II., D.R.J. Chillingworth

---Quadratic forms and the Birman-Craggs homomorphisms, Dennis Johnson

---The Chillingworth class is a signed stable length, Ingrid Irmer

---The mapping class group action on the homology of the configuration spaces of surfaces, Tetsuhiro Moriyama

---The μ-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented, 2-manifold, Joan Birman and R. Craggs

---Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I, Shigeyuki Morita

---A primer on handlebody groups, Sebastian Hensel

---The structure of the Torelli group II: a characterization of the group generated by Dehn twists on bounding curves, Dennis Johnson

---The Johnson homomorphism and its kernel, Andrew Putman

---The structure of the Torelli group III: the abelianization of I, Dennis Johnson

---Conjugacy relations in subgroups of the mapping class group and a group-theoretic description of the Rochlin invariant, Dennis Johnson

---Notes on the Sigma invariants, Version 2, Ralph Strebel

---On the geometry and dynamics of diffeomorphisms of surfaces, William Thurston

--- The lower central series and pseudo-Anosov dilatations, Benson Farb, Chris Leininger, and Dan Margalit

---Normal generators for mapping class groups are abundant, Justin Lanier and Dan Margalit

---On finite generation of the Johnson filtrations, Thomas Church, Mikhail Ershov, Andy Putman

---A geometric invariant of discrete groups, Robert Bieri, Walter Neumann, and Ralph Strebel

---Fibering rigidity of 3-manifolds with Torelli monodromy, Ingrid Irmer