Kirby Problem 4.79

Definitions and mathematical statement

Let $X$ be a closed simply connected smooth four-manifold and set $X^\circ=X\setminus\operatorname{int}(B^4)$. The group $D_X=\operatorname{Diff}_\partial(X^\circ)$ consists of diffeomorphisms equal to the identity near the boundary, with its $C^\infty$ topology and identity basepoint. The corresponding homeomorphism group has the compact-open topology. Inclusion gives \[ i_*:\pi_k(D_X)\to\pi_k\operatorname{Homeo}_\partial(X^\circ). \] Using a fixed boundary collar to attach a punctured $S^2\times S^2$, extension by the identity defines a one-step stabilization map \[ s_*:\pi_k(D_X)\to \pi_k\operatorname{Diff}_\partial(X^\circ\#(S^2\times S^2)). \]

For every integer $k>0$, do there exist such an $X$ and a class $\alpha\in\pi_k(D_X)$ with $i_*(\alpha)=0$ but $s_*(\alpha)\ne0$? In particular, $\alpha$ itself must be nonzero. The four-manifold may depend on $k$.

A class here is a based continuous $S^k$-family of diffeomorphisms. The first condition means it extends over a $(k+1)$-ball as a family of boundary-fixed homeomorphisms. The second says no such extension by diffeomorphisms exists even after one stabilization. These are conditions on the whole parametrized family, not on the isotopy class of each individual member considered separately. It does not require persistence after every finite stabilization, which is a stronger property. Exhibiting a nontrivial mapping class at $k=0$ would not answer the requested higher-dimensional family question.

Short English statement

In every positive parameter dimension, can an exotic diffeomorphism family remain nontrivial after one stabilization?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2955 (KP-4.79), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.79. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.