Georgia Institute of Technology College of Sciences

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.

Two prominent questions in low dimensional topology are: which knots are slice, and which $\mathbb{Q}$-homology $S^3$'s bound $\mathbb{Q}$-homology $B^4$'s? These questions are connected by a theorem that states if a knot $K$ in $S^3$ is slice, then the 2-fold branch cover of $S^3$ over $K$ bounds a $\mathbb{Q}$-homology $B^4$. In this talk we introduce a generalization of $\chi$-sliceness of links to the rational homology context, generalize the earlier theorem to state that for a rationally $\chi$-slice link $L$, for all sufficiently large primes $p$, the $p$-fold cyclic branch cover of $S^3$ over $L$ bounds a $\mathbb{Q}$-homology $B^4$, and examine a connection to a number-theoretic obstruction on the Alexander polynomial.