Reflection conjugation versus inversion for the 4-ball mapping class group

Determine whether conjugation by a reflection equals the inversion map in the mapping class group \(\pi_0\mathrm{Diff}(B^4,\partial B^4)\).

Background

The paper studies the Davis reflection construction, which associates to a compact manifold-with-boundary $X$ a closed quotient $Q(X)$ and induces a homomorphism from the relative diffeomorphism mapping class group of $X$ to that of $Q(X)$. In dimensions other than four, the authors show that conjugation by a linear reflection on $B^d$ acts as inversion on $\pi_0\mathrm{Diff}(B^d,\partial B^d)$, allowing local diffeomorphisms to be controlled in the quotient construction.

The unresolved four-dimensional case concerns whether the same identity holds for $\pi_0\mathrm{Diff}(B^4,\partial B^4)$. Resolving it would determine whether the argument used to prove triviality of the image of local diffeomorphisms, and hence the corresponding injectivity result for the Davis quotient when the subgroup is not orientation-preserving, extends to dimension four.