Math 8803 Torelli groups Spring 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: 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

Homework suggestions

• January
  1. Summarize one of the proofs that the mapping class group is generated by Dehn twists
  2. Find a minimal generating set for the mapping class group of a punctured surface
  3. Perform the euclidean algorithm for curves with several specific elements of H_1(S_g;Z)
  4. Determine which pairs of vectors v and w satisfy t_v+w=t_v+t_w
  5. What is the analogue of the symlpectic group for a surface with boundary?
  6. Classify bounding pair maps up to conjugacy in the Torelli group
  7. Complete the third proof that the Torelli group is torsion free
  8. Construct n-gons in the complex of cycles. Which integral polygons can be realized? Solution by Tao Yu

• February
  1. Complete one of the 5 calculations required to complete Johnson I.
  2. Explain the Rochlin invariant.
  3. Show that the BCJ maps are homomorphisms.
  4. Compute the image of a BP map under a BCJ map.
  5. Find generators for the kernel of the Chillingworth map.
  6. Exposit Irmer's paper. Exposition and new proof by Tao Yu
  7. Exposit Moriyama's paper for the case n=2.

• March / April
  1. Explain Morita's argument for the image of handlebody Torelli under Johnson
  2. Check that last relation in Johnson II, to show that K/T is abelian
  3. Explain (part of) Gaifullin's new paper on the homology of Torelli
  4. Explain the definition of the Casson invariant for a 3-manifold
  5. Prove that the abelianization is generated by twists of genus one
  6. Explain Irmer's theorem that Torelli monodromies for fibered 3-manifolds are unique
  7. Prove the BNS theorem
  8. Relate the BNS invariant to actions on R-trees
  9. Explain the proof of the general result of Church-Ershov-Putman
  10. Prove that K is finitely generated without the Zariski topology
  11. Explain why the cohomological dimension of Torelli is 3g-5

---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