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Triviality of the compactly-supported mapping class group of ℝ⁴

Determine whether the compactly-supported mapping class group MCG_c(ℝ⁴) is trivial.

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Background

The authors recall Watanabe’s result that Diff(D⁴, ∂) is not contractible, disproving the 4D Smale conjecture, and then turn to the compactly-supported mapping class group of ℝ⁴. They emphasize that MCG_c(ℝ⁴) ≅ MCG(D⁴) ≅ MCG(S⁴) and pose the question of its triviality.

This question motivates their construction of exotic ℝ⁴ with nontrivial compactly-supported mapping class groups.

References

Watanabe has proved that $\mathrm{Diff}(D4, \partial)$ is not contractible (thus disproving the four-dimensional Smale conjecture), but the following important question remains open: is $\MCG_{c}(\mathbb{R}4)$ trivial?

— On four-dimensional Dehn twists and Milnor fibrations (2409.11961 - Konno et al., 18 Sep 2024) in Introduction, Subsection “Exotic Dehn twists on open, contractible, and closed manifolds”