What is the homotopy type of the group of diffeomorphisms of the 4-sphere?

Definitions and mathematical statement

Let $D^4=\{x\in\mathbb R^4:\|x\|\le1\}$, and let $\mathcal D=\operatorname{Diff}(D^4\operatorname{ rel}\partial D^4)$ be the group of smooth diffeomorphisms fixing the boundary pointwise. Give it the $C^\infty$ topology: convergence means uniform convergence in local charts of every derivative on compact sets, and use the identity diffeomorphism as base point. This is the boundary-fixing convention in the source; collar-fixing models must be compared explicitly when used.

The computational targets are \[ \pi_j(\mathcal D,\mathrm{id})\quad(j\ge1), \qquad \pi_0(\mathcal D),\qquad H_j(\mathcal D;\mathbb Z). \] Here $\pi_j$ consists of based homotopy classes of maps from $S^j$, $\pi_0$ records isotopy components, and $H_j$ denotes singular homology. The source asks for further determinations of these groups as steps toward the homotopy type of $\operatorname{Diff}(S^4)$, through its stated relation to $O(5)\times\mathcal D$, with $O(5)$ the orthogonal matrix group.

This is a computation program, rather than a claim that every group displayed above vanishes. Specify the degree, coefficients, and whether a calculation concerns the identity component or the whole space. Finding the homotopy type would organize all these invariants; an isolated nonzero class alone would not determine the complete type.

Short English statement

Determine further homotopy and homology groups of the boundary-fixing diffeomorphism group of the four-ball, and hence understand the diffeomorphism space of the four-sphere.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3417 (OPG-37159), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, What is the homotopy type of the group of diffeomorphisms of the 4-sphere?. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/what_is_the_homotopy_type_of_the_group_of_diffeomorphisms_of_the_4_sphere.