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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.

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    $\begingroup$ 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. $\endgroup$ – Marco Golla Jan 7 at 21:50
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    $\begingroup$ 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. $\endgroup$ – Kyle Hayden Jan 9 at 2:29

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