Monday

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).