Is anything known about the set of concordance classes (also called pseudoisotopy classes) of the relative to the boundary diffeomorphisms of $D^4$?



Concordance classes of diffeomorphisms of $D^4$
It is the trivial group.


Firstly, by isotopy extension the extension-by-the-identity homomorphism to the group of pseudoisotopy classes of orientation-preserving diffeomorphisms of $S^4$,
$$\widetilde{\pi}\_0(\mathrm{Diff}\_\partial(D^4)) \to \widetilde{\pi}\_0(\mathrm{Diff}^+(S^4)),$$
is surjective with kernel generated by the Dehn twist around the boundary. But the Dehn twist is isotopic to the identity for $D^4$ (it is easy to produce an explicit isotopy), so this homomorphism is an isomorphism.


Then for $S^4$ you just invoke Theorem 1 of *[Kreck's Isotopy classes of diffeomorphisms of $(k-1)$-connected almost-parallelizable $2k$-manifolds](https://link.springer.com/chapter/10.1007%2FBFb0088108)*, which says that for any closed 1-connected smooth $4$-manifold $M$, the homomorphism
$$\widetilde{\pi}\_0(\mathrm{Diff}(M)) \to \mathrm{Aut}(H\_2(M;\mathbb{Z})),$$
induced by the action on homology, is injective.


