Kirby Problem 4.73

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\operatorname{Diff}_{\partial}(D^4)$ be the topological group of smooth diffeomorphisms of the closed disk that are the identity on a neighborhood of its boundary, with the smooth topology; $B\operatorname{Diff}_{\partial}(D^4)$ is its classifying space. Let $\operatorname{Top}(4)$ be the topological group of homeomorphisms of $\mathbb R^4$ fixing the origin, with the compact-open topology, and let $O(4)$ be the orthogonal group. Use \[ F_4=\operatorname{hofib}\left(BO(4)\longrightarrow B\operatorname{Top}(4)\right) \] for the based homotopy fiber. This is the space denoted $\operatorname{Top}(4)/O(4)$ in smoothing theory. Write $\Omega^4_0F_4$ for the path component of the constant map in the based fourfold loop space $\operatorname{Map}_*(S^4,F_4)$. A weak homotopy equivalence induces a bijection on path components and isomorphisms of all based homotopy groups.

Statement recorded in the supplied source.

Does the four-dimensional analogue of Morlet's equivalence hold: \[ B\operatorname{Diff}_{\partial}(D^4)\simeq\Omega^4_0\left(\operatorname{Top}(4)/O(4)\right) \] as a weak homotopy equivalence? The disk dimension is four and the loop-space component is the component of the constant map. [S2]

Short English statement

Does Morlet’s comparison between smooth disk diffeomorphisms and topological smoothing data remain a weak equivalence for the four-disk?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2949 (KP-4.73). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.73 and Remarks, PDF page 250. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.