REU: Mapping Class Groups of Surfaces (PAGE UNDER CONSTRUCTION)

Cornell University, June 6 - July 29, 2005

Supervised by Tara E. Brendle

Graduate Assistant: Heather Armstrong

Undergraduate Participants:

  • Tova Brown, UC Santa Cruz
  • Tom Church, Cornell University
  • Peter Maceli, Cornell University
  • Aaron Pixton, Princeton University
  • Vijay Ravikumar, Amherst College

    Project Summaries

    Background: Mapping Class Groups and the Torelli Group

    We denote a compact, orientable surface with genus g and b boundary components by Sg,b. For our purposes we generally either take b to be 0 or 1. For such a surface we may define the mapping class group, Mod(Sg,b), as the group of homeomorphisms from the surface to itself which fix the boundary pointwise, modulo isotopy. Mod(Sg,b) is generated by a certain type of homeomorphism called a Dehn twist. Given a simple closed curve c on Sg,b, the Dehn twist about c, denoted Tc, is obtained by cutting the surface along c, twisting the surface on one side of the cut, and then reattaching along the cut. Sg,b may be equipped with a symplectic basis for its first homology, with intersection number as the associated bilinear form. Since homeomorphisms from the surface to itself are bijective and preserve intersection number, all elements of Mod(Sg,b) map the symplectic basis to another symplectic basis. Thus there is a natural homomorphism from Mod(Sg,b) to the group of symplectic automorphisms on H1(Sg,b, Z). The kernel of this homomorphism, consisting of those isotopy classes of homeomorphisms which act trivially on homology, is called the Torelli group, and denoted Ig,b. The Torelli group is generated by elements of the form TcT-1d, where c and d are disjoint (and distinct) homologous simple closed curves. Ig,b also contains all Dehn twists about separating curves. In fact, the group generated by Dehn twists about separating curves is an important normal subgroup of the Torelli group, known as the Johnson kernel and denoted Kg,b.

    The Birman-Craggs-Johnson Homomorphism

    One way to begin understanding a complicated group like the Torelli group is to find a homomorphism which maps the group onto a group you do understand and examine what structure is preserved. In the late 1970's, Birman-Craggs used the Rochlin invariant of homology 3-spheres to construct a large finite class of homomorphisms from the Torelli group I onto Z/2Z. These were the first abelian quotients of the Torelli group. Soon after, Dennis Johnson repackaged these as a single homomorphism σ: I → B3, where B3 is a certain F2-vector space of Boolean (square-free) polynomials. The kernel of the Birman-Craggs-Johnson homomorphism σ is equal to the intersection of the kernels of all the Birman-Craggs homomorphism.

    The BCJ Map and the Homology of the Torelli Group

    Although Johnson computed the abelianization of the Torelli group in the early 1980's, the higher-dimensional homology of the Torelli group remains poorly understood. In fact, it is an open question whether the Torelli group is finitely presentable (which would imply that H2(I) is finitely generated). In this project, we explore the induced homomorphism σ*: H2(I,F2) → H2( B3,F2). Brendle-Farb have used abelian cycles to construct nontrivial homology classes in H2(I,F2). Their work focused on the Johnson kernel and used abelian cycles corresponding to Dehn twists about disjoint separating curves. Here we construct abelian cycles from bounding pair maps as well. In doing so, we improve their lower bound on the rank of H2(I,F2) from a degree 4 polynomial in the genus g to a degree 6 polynomial. In addition, we examine the subgroup W of H2( B3,F2) generated by the image of all abelian cycles under σ* and prove that W does not contain a certain class of elements of H2( B3,F2). For more information about this project, click here.
    Click here to see Vijay's abelian cycle construction that proves the degree 6 lower bound.

    A new topological interpretation of the kernel of the BCJ map

    We propose a possible topological characterisation of the intersection of the Johnson kernel and the Birman-Craggs-Johnson kernel, where the Johnson kernel is the group generated by Dehn twists about separating curves. To begin with, we consider three types of operations on compositions of Dehn twists about separating curves, all of which preserve the image of those twists under the Birman-Craggs-Johnson homomorphism. We observe that, at least in some cases, a finite sequence of these moves produces a Dehn twist about a trivial curve, demonstrating that the original composition of Dehn twists is in the BCJ kernel. We suggest that this process may be possible for any element of both the Johnson kernel and the BCJ kernel. We give a partial proof in the case of S2,1. In order to complete the proof in this case, it suffices to show that two Dehn twists about separating curves have the same image if and only if the curves bound surfaces with homologous bases.

    The Magnus Representation of the Torelli Group and Higher Intersection Forms

    Given a surface Sg,1 with one boundary component, choose a basepoint b on the boundary; then the fundamental group Γ1=π1(Sg,1, b) will be free on 2g generators. The Magnus representation of the Torelli group was first defined using Fox calculus via the action of Ig,1 on Γ1. We can also define the Magnus representation topologically. Let S~ be the universal abelian cover of Sg,1; in other words, S~ is the regular covering space corresponding to the commutator subgroup of Γ1. An element f of Ig,1 can be represented by a homeomorphism of Sg,1, and then f lifts to a homeomorphism f~ of S~. The action of the lifted homeomorphism f~ on the relative homology H1(S~,p-1(b))≅Z[H]2g defines the Magnus representation r2:Ig,1→GL2g(Z[H]).

    In analogy with the algebraic intersection form defined on H1(Sg,1), we define a ``higher'' intersection form on H1(S~,p-1(b)), the homology of the abelian cover. This higher intersection form allows us to generalize a theorem of Suzuki by describing all the relations between the images of two twists around separating curves.

    A series of Magnus-like representations can be defined corresponding to the terms of the lower central series of Γ1. For each term Γk in the lower central series, the Johnson filtration M(k) of Mod(Sg,1) is the subgroup of Mod(Sg,1) consisting of homeomorphisms that act trivially on Γ1/Γk. For each k, let S(k) be the regular covering space of Sg,1 corresponding to Γk. Given a homeomorphism f in M(k), f lifts to a homeomorphism f~ of S(k), which acts on the homology of S(k). This action defines rk:M(k)→GL2g(Z[Γ1/Γk]), the Magnus representation associated to Γk. These representations are closely related to the Johnson homomorphisms τk:M(k)→ Hom(H, Γk/Γk+1). We show that every Magnus representation rk lifts a significant number of the Johnson homomorphisms. We also exhibit a new abelian quotient of the Johnson kernel Kg,1=M(3).