Concordance of surfaces

Abstract: Concordance of surfaces in 4-manifolds is very different from concordance of knots in 3-manifolds – in particular, there are fewer obstructions. Recently, the Freedman-Quinn invariant fq(R,S), which is defined for certain (based-)homotopic surfaces R,S and is a concordance obstruction, has appeared in several theorems. In this talk, I will describe an invariant km(R,S) due to Stong, which is defined when fq(R,S) vanishes. The km invariant also obstructs concordance, but is subtle in that

(1) It is hard (or at least annoying) to define km,

(2) It is hard (read: a source of eternal angst) to construct examples achieving given km values,

(3) Some (overly-)convenient hypotheses that are often made in 4D topology cause km to automatically vanish.

Klug and I previously showed how to construct pairs of 2-spheres R,S with km(R,S) non-vanishing (thus proving the necessity of hypotheses for some other theorems). In this talk, I’ll define these various terms (sort of), give a lot of motivation for why concordance of surfaces is interesting (by first explaining why it isn't), tell you some theorems about concordance of surfaces (which is actually very interesting) and a bit about ongoing work. 

(This is joint work with Michael Klug.)