Georgia Institute of Technology College of Sciences

Heegaard Floer homology is a modern algebraic package that has seen utility in many areas of low dimensional topology. In this introductory talk, we will give a birds eye view of its construction and properties, and present a highlight reel on the problems it helps solve. As many of our beloved faculty find use for Floer homology in their research, this talk intends to ease the newcomer into the vernacular they may hear in a Seminar talk, and at the same time, convince them that it is a beautiful and rich field of study. We do not assume knowledge in low dimensional topology. It is also rumoured that some Dunkin Donuts will be in attendance.

In his 1956 paper "On manifolds homeomorphic to the 7-sphere'', John Milnor constructed some examples of manifolds that are homeomorphic, but not diffeomorphic, to the standard unit sphere. They are now called exotic 7-spheres. This example established that the differential structure of a manifold can carry information not given by its topological structure. Thus, Milnor founded differential topology as a stand-alone field. On my first reading of the paper, I thought that many of the choices Milnor made on his road to constructing the first exotic spheres seemed rather strange and arbitrary. Why 7-spheres? Why look for them among $S^3$-bundles over $S^4$ with structure group $SO(4)$? And what's the motivation behind his complicated mod 7-valued lambda invariant that detects exotica in these examples? Fortunately, Milnor answered some of these questions in his essay "Classification of $(n-1)$-connected $2n$-dimensional manifolds and the discovery of exotic spheres''. This talk is an attempt to understand this essay. 

Modern homology theories have given many knot invariants with the following useful properties: they are additive with respect to connected sum, they give a lower bound for a knot's slice genus, and this lower bound is equal to the slice genus for torus knots. These invariants, called slice-torus invariants, include the Ozsváth–Szabó $\tau$ and Rasmussen $s$ invariants. We discuss how, on a large class of knots, the value of a slice-torus invariant is fully determined by these properties, and can be computed without reference to the homology theory. We also discuss results that follow from the existence of slice-torus invariants, and a potential connection to the smooth 4-dimensional Poincaré conjecture.

The classical Laudenbach-Poénaru theorem states that any diffeomorphism of $\#_n S^1 \times S^2$ extends over the boundary connect sum of $n$ $S^1 \times B^3$'s. This implies the familiar fact that in Kirby diagrams for closed 4 manifolds, you do not need to specify the attaching spheres for 3 handles; it is also the backbone result of trisection theory, which allows one to describe a closed 4 manifold by three cut systems of curves on a surface. We extend this result to the case of infinite 4-dimensional 1-handlebodies, with an eye towards developing trisections for noncompact 4 manifolds. The proof is geometric and based on extending the recent proof of Laudenbach-Poenaru due to Meier and Scott.