Math 8803Torelli groupsSpring 2018 


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: BirmanCraggsJohnson 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, Nonfiniteness of cohomology in genus 2 and genus 3, Akita's nonfiniteness, cohomological dimension
Representation stability: Parameterized AbelJacobi maps, Generating the Johnson filtration, finite generation of second homology as an Spmodule
Week  Dates  Topics  Reading  Lectures  Notes 
1  Jan 812  Overview / MCG  Primer  Intro MCG 1  No class Mon ("ice storm") 
2  Jan 1519  MCG  Primer  MCG 2  MLK day + snow day 
3  Jan 2226  Sp rep  Primer  Sp rep  
4  Jan 29Feb 2  Complex of cycles  HatcherM BBM Slides 
Torelli gens  
5  Feb 5  Feb 9  Generating Torelli  HatcherM  
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 
BCJ  
9  Mar 5  Mar 9  BCJ / ZHS^{3}  Morita I  Homology 3spheres  
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  pseudoAnosovs / Torelli space  
13  Apr 16  Apr 20  K_{g} is fg  ChurchErshovPutman  K_{g}  
14  Apr 23  Problems  Problem list  Problems  
Course notes 
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, KaiUwe 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 BirmanCraggs 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 μinvariant of 3manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented, 2manifold, Joan Birman and R. Craggs
Casson's invariant for homology 3spheres 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 grouptheoretic 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 pseudoAnosov 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 3manifolds with Torelli monodromy, Ingrid Irmer