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Lei Chen



5. The universal surface bundle over the Torelli space has no sections. Arxiv
For g>3, we give two proofs of the fact that the Birman exact sequence for the Torelli group does not split.
We also prove that the universal n-pointed Torelli bundle only has n sections up to homotopy. (2017)
4. Section problems for configuration spaces of surfaces. Arxiv
In this paper we give a close-to-sharp answer to the basic questions:
When is there a continuous way to add a point to a configuration of n ordered points on a surface S of finite type so that all the points are still distinct?
We will answer the question for R^2, S^2 and S_g when g>1.(2017)
3. Surjective homomorphisms between surface braid groups. Arxiv Notes
Classify surjective homomorphisms from PB_n(S_g,p) to PB_m(S_g,p), n-strand braid group to m-strand braid group.
Then we compute the automorphism group of PB_n(S_g,p)
Surprisingly, in contrast to the n=1 case, any automorphism of PB_n(S_g,p), n>1 is geometric.(2017)
2. The number of fiberings of a surface bundle over a surface. Arxiv
Compute the number of ways Atiyah Kodaira example can fiber over a surface.
We also compute the fibering number of some other examples. (2017)
1. The universal n-pointed surface bundle only has n sections. Arxiv
Classify homomorphisms from PB_n(S_g) to Pi_1(S_g), n-strand braid group to 1-strand braid group.
Using this, we classify sections of the universal n-pointed surface bundle and the universal n-pointed hyper-elliptic surface bundle. (2016)
Accepted by Journal of Topology and Analysis.

Not for publication

In Winter 2014 I passed my topic exam, on Characteristic Classes of Surface Bundles, under the supervision of Benson Farb. The proposal is here.



2. ICMS Edinburgh
Slides: Surjective homomorphisms between surface braid groups.
1. Oberwalfach surface bundle
Youtube video: The section problems.
Notes: The section problems


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Mapping Class Groups

  • Burau Representation PDF
  • Describe Johnson homomorphism using Massay Product PDF
  • Homology 3-sphere and Casson Invariant PDF
  • Torsions in mapping class group PDF
  • 2017.7.7: Birman-Hilden Theorem PDF
  • 2017.7.9: counting short geodesics on flat torus PDF
  • 2017.7.24: Earle-Eells Theorem PDF
  • 2017.7.25: Cohomology of Braid group with Z2 coefficient PDF
  • Random Topics

  • Finite generated residually finite group is Hopfian PDF
  • Grothendieck Riemann Roch PDF
  • polynomial growth--nilpotent group PDF
  • Reeb Stability Thurston PDF
  • Reading Groups

  • Reading group on moduli space Spring 2017 Website
  • About me

    I'm a graduate student in at The University of Chicago, interested in Mapping Class Group and topology in general.

    Postal address
    5734 S. University Avenue
    Chicago, IL 60637-1514
    Office E105 in Eckhart
    Advisor Benson Farb
    Here is my CV