Narrow Results

We continue to study the algebraic property of the linear representation of the mapping class group of a closed oriented surface of genus 2 constructed by V. F. R. Jones. We consider the perturbation of the representation at the involved parameter t=1. This perturbation naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients. Our arguments follow the same line of our previous paper which dealt with the perturbation at t=-1. However, the obtained results may still suggest a new aspect of the representation.

We observe that the determinant of the representation provides a little restriction for the structure of the graded quotients introduced in both [2] and [3] that any one of them does not contain the trivial 1-dimensional summand.

The Torelli group, , is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying : bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on , while SIP-maps have received relatively little attention until recently, due to an infinite presentation of introduced by Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman–Craggs–Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP(S_g), and show it is an infinite index subgroup of .

In this paper we prove that the Rohlin invariant is the unique invariant inducing a homomorphism on the Torelli group. Using this result we generalize the construction of invariants of homology 3-spheres from families of trivial 2-cocycles on the Torelli group given by Pitsch to include invariants with values on an abelian group with 2-torsion.

The Reshetikhin-Turaev representation of the mapping class group of an orientable surface is computed explicitly in the case r = 4. It is then shown that the restriction of this representation to the Torelli group is equal to the sum of the Birman-Craggs homomorphisms. The proof makes use of an explicit correspondence between the basis vectors of the representation space, and the Z/2Z-quadratic forms on the first homology of the surface. This result corresponds to the fact, shown by Kirby and Melvin, that the three-manifold invariant when r = 4 is related to spin structures on the associated four-manifold.

The quantum group construction of Reshetikhin and Turaev provides representations of the mapping class group, indexed by an integer parameter r. This paper presents computations of these representations when r=6, and analyzes their relationship to other topological invariants. It is shown that in genus 2, the representation splits into two summands. The first summand factors through the mapping class group action on the first homology of the surface with Z/3Z coefficients, while the second summand can be analyzed via its restriction to the subgroup of the mapping class group which is normally generated by the sixth power of a Dehn twist on a nonseparating curve. This analysis reveals a connection to the homology intersection pairing on the surface, and also yields information about the kernel and image of the representation. It is also shown that the representation yields a family of 2-dimensional nonabelian representations of the Torelli group.

This paper continues the program established by the author in [Wr] to relate the Reshetikhin-Turaev representations at specific roots of unity to classical invariants.

Suppose S is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, , of S; a complex closely related to complexes studied by Bestvina–Bux–Margalit and Hatcher. A path in corresponds to a homotopy class of immersed surfaces in S × I. This observation is used to devise a simple algorithm for constructing quasi-geodesics connecting any two vertices in , and for constructing minimal genus surfaces in S × I. It is proven that for g ≥ 3 the best possible bound on the distance between two vertices in depends linearly on their intersection number, in contrast to the logarithmic bound obtained in the complex of curves. For g ≥ 4 it is shown that is not δ-hyperbolic.

We show that the Torelli group of a closed surface of genus $\ge 3$ acts nontrivially on the rational cohomology of the space of $3$-element subsets of that surface. This implies that for a Riemann surface of genus $\ge 3$, the mixed Hodge structure on the space of its positive, reduced divisors of degree $3$ does in general not split over $\mathbb{Q}$.