Absolutely exotic contractible four-manifolds
Since exotic four-manifolds are always "in", I wanted to talk about a cool construction of exotica I like due to Akbulut-Ruberman. In their paper, they construct compact contractible four-manifolds which are "absolutely exotic", which means they are homeomorphic, but there is no diffeomorphism between them. To clarify the terminology, often times, for manifolds with boundary, one talks about "relatively exotic" manifolds, which means that you have a homeomorphism from one four-manifold to another, but the boundary homeomorphism doesn't extend to a diffeomorphism of the four-manifolds. [For context, it's worth pointing out that long before, we know how to construct lots of non-compact, contractible exotica: namely exotic $\mathbb{R}^4$'s. In fact, Taubes proved there are uncountably many exotic $\mathbb{R}^4$'s.] I want to describe their work (but with a bit of a Floer homology lean). For simplicity, let's construct a pair of n...