Puzzles about trisections of 4-manifolds

A trisection of a smooth 4-manifold is a very natural kind of decomposition into three elementary pieces which I will describe. Trisections are a natural 4-dimensional analogue of Heegaard splittings of 3-manifolds, a class of decompositions into two pieces that have yielded tremendous insight into 3-dimensional topology, so the philosophy is that trisections should give a way to port 3-dimensional techniques, questions and results to dimension four. I will describe all this and then explicitly describe a strange local operation on trisections that does not seem to have a 3-dimensional analogue, and about which I know very little. The main puzzle is: Does this operation really change trisections or not? Does it perhaps change some trisections and not others? If so, what does that tell us about 4-manifolds?

For non-topologists in the audience the goal should be to have fun with 4-dimensional visualization; I will not assume a strong topology background and will focus on the pictures.