# Are there exotic twisted doubles of 4-manifolds?

Take a smooth 4-manifold $$X$$ whose boundary has a diffeomorphism $$\tau: \partial X \to \partial X$$ that extends to a homeomorphism but not a diffeomorphism of $$X$$. (By Matveyev and Curtis-Freedman-Hsiang-Stong, any two homeomorphic smooth 4-manifolds are related by cutting out such an $$X$$ and regluing it using $$\tau$$.) The "twisted double" $$DX_\tau= X \cup_\tau -X$$ obtained by gluing $$X$$ to itself via $$\tau$$ is then homeomorphic to the regular double, $$DX=X \cup_{\operatorname{id}} - X$$. In all examples I know of, the twisted double is actually diffeomorphic to the regular double. (An example is nicely illustrated in Figure 2 of this paper.)

Are there known examples of exotic twisted doubles of 4-manifolds?

I'm also curious about the analogous question in higher dimensions.

• Nice question! In dimension $d \ge 6$, every exotic $S^d$ is (in your language) a twisted double of the ball; this was proven by Smale. – Marco Golla Jan 7 at 21:50
• Thanks, Marco! The Smale reference led me to Ranicki’s book “High Dimensional Knot Theory”, which has a nice discussion. Apparently work of Smale and Barden shows that all simply-connected manifolds of odd dimension $d\geq 5$ are twisted doubles. – Kyle Hayden Jan 9 at 2:29