Tuesday

If Y is a closed oriented 3-manifold, its Chern-Simons function is a function on a certain infinite-dimensional space, and the instanton Floer homology I_*(Y) is constructed as the Morse homology of this function. What’s special about the Chern-Simons function is that it depends only on the topology of Y, not any other geometric input or auxiliary data. Using the corresponding filtration, the irreducible instanton homology of Y can be given the structure of a persistence module, and from there one may extract numerical invariants of Y.

I will discuss how this idea, combined with Floer's exact triangle and an exact triangle due in the admissible case to Culler--Daemi--Xie, leads to a proof of the cosmetic surgery conjecture for knots in S^3 and surgery slope 1/n. This leaves open only the possibility S^3_2(K) ~ S^3_-2(K).

In this talk we show that for any smooth 4-manifold X homeomorphic to a Gompf nucleus N(2n) and any odd prime p, the standard Z/p-action given by rotation in the fibers of the Seifert-fibered boundary cannot extend smoothly to a Z/p-action over X (with one exceptional case), whereas in some cases these actions do extend topologically. In particular, we show that for each prime p>=5 and each n>=1 there exists a non-smoothable Z/p-action on N(2pn) extending the standard Z/p-action on its boundary. The proof makes use of invariants coming from Seiberg-Witten Floer K-Theory as well as some equivariant index theory.

In the 1970’s dihedral representations of knot groups were used to define twisted signature-type invariants which generalize the older invariants of Levine and Tristram. The most prominent examples are the Casson-Gordon invariants, which provide obstructions to being topologically slice as well as more sensitive obstructions to being ribbon. At the same time, Cappell and Shaneson observed a similar obstruction implicit in a formula for the Rokhlin invariant of a 3-manifold presented as an irregular dihedral cover. More recently, Kjuchkova formulated this observation
into an invariant of Fox-p-colored knots which obstructs the knot being ribbon. We will introduce a similar invariant based on the same topological setup, but which is extremely computable, and also obstructs a knot from being ribbon. We will demonstrate how to compute it for some surprisingly large examples, and then we will survey some constructions in the literature of potential counter-examples of the Slice-Ribbon Conjecture. This is joint work with Sylvain Cappell.

We discuss how the moduli space of SO(3) monopoles can be used to define relations between some invariants of four-manifolds with b+ even.

I will survey a joint project with Dave Auckly in which we show that homological invariants of the diffeomorphism group of a smooth 4-manifold can be infinitely generated. This has similar consequences for the algebraic topology of the space of embeddings of surfaces and 3-manifolds in 4-manifolds, and spaces of positive scalar curvature on 4-manifolds.