A handle-holding approach to Wall-type stabilization problems

Kyle Hayden (Rutgers)

In dimension four, the differences between continuous and differential topology are significant but fundamentally unstable. A longstanding question due to Wall aims to quantify this instability. I will begin by introducing Wall's stabilization problem (plus some of its friends) and highlight recent breakthroughs proven via Floer homology. Then I will sketch a new handle-theoretic approach to these problems. As a proof of concept, I will close by sketching how this approach furnishes new counterexamples to an analog of Wall's problem for knotted surfaces. The key obstruction distinguishing these surfaces comes from the "universal" version of Khovanov homology.