Kirby Problem 4.78

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 $X$ be a closed simply connected smooth four-manifold, choose a smooth embedded four-ball, and put $X^{\circ}=X\setminus\operatorname{int}(D^4)$. For $N\ge0$ set \[ M_N=X^{\circ}\#N(S^2\times S^2),\qquad G_N=\operatorname{Diff}_{\partial}(M_N). \] Here $\operatorname{Diff}_{\partial}$ has the smooth topology and consists of diffeomorphisms that are the identity near the boundary. Form each connected sum in a newly attached boundary collar, so extension by the identity defines a stabilization homomorphism $s_N:G_N\to G_{N+1}$. Let $BG_N$ be the classifying space, and let $H_k(-;\mathbb Q)$ denote singular homology with rational coefficients.

Statement recorded in the supplied source.

For every $X$ as above and every $k\ge0$, does there exist $N_0=N_0(X,k)$ such that for every $N\ge N_0$ the stabilization map induces an isomorphism \[ (Bs_N)_*:H_k(BG_N;\mathbb Q)\longrightarrow H_k(BG_{N+1};\mathbb Q)? \] The original source specifies the boundary-fixed group of the punctured manifold; this is the intended meaning of its homological-stability question. [S2]

Short English statement

Do the rational homology groups of the classifying spaces of boundary-fixed diffeomorphisms eventually stabilize when a punctured simply connected four-manifold is repeatedly summed with the product of two spheres?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2954 (KP-4.78). 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 4.78 and Remarks, PDF page 255. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.