Georgia Institute of Technology College of Sciences

By a classical result of Eliashberg, contact manifolds in dimension 3 come in two flavors: tight (rigid) and overtwisted (flexible). Characterized by the presence of an "overtwisted disk", the overtwisted contact structures form a class where isotopy and homotopy classifications are equivalent.In higher dimensions, a class of flexible contact structures is yet to be found. However, some attempts to generalize the notion of an overtwisted disk have been made. One such object is a "plastikstufe" introduced by Niederkruger following some ideas of Gromov. We show that under certain conditions, non-isotopic contact structures become isotopic after connect-summing with a contact sphere containing a plastikstufe. This is a small step towards finding flexibility in higher dimensions. (Joint with E. Murphy, K. Niederkruger, and A. Stipsicz.)

It is a theorem of Bass, Lazard, and Serre, and, independently, Mennicke, that the special linear group SL(n,Z) enjoys the congruence subgroup property when n is at least 3. This property is most quickly described by saying that the profinite completion of the special linear group injects into the special linear group of the profinite completion of Z. There is a natural analog of this property for mapping class groups of surfaces. Namely, one may ask if the profinite completion of the mapping class group embeds in the outer automorphism group of the profinite completion of the surface group. M. Boggi has a program to establish this property for mapping class groups, which couches things in geometric terms, reducing the conjecture to determining the homotopy type of a certain space. I'll discuss what's known, and what's needed to continue his attack.

The hyperelliptic Torelli group SI(S) is the subgroup of the mapping class group of a surface S consisting of elements which commute with a fixed hyperelliptic involution and which act trivially on homology. The group SI(S) appears in a variety of settings, for example in the context of the period mapping on the Torelli space of a Riemann surface and also as a kernel of the classical Burau representation of the braid group. We will show that the cohomological dimension of SI(S) is g-1; this result fits nicely into a pattern with other subgroups of the mapping class group, particularly those of the Johnson filtration. This is joint work with Leah Childers and Dan Margalit.

I will talk briefly about how the study of fibred knots and Thurston's classification of automorphisms of surfaces in the 70's lead to Gabai and Oertel's work on essential laminations in the 80's. Some of this structure, for instance fractional Dehn twist coefficients, has implications in contact topology. I will describe results and examples, both old and new, that emphasize the special nature of S^3. This talk is based on joint work with Rachel Roberts.