Kirby Problem 4.14

Definitions and mathematical statement

Use the source's broad cork convention. A cork $(C,f)$ consists of a compact contractible smooth $4$-manifold $C$ and a boundary diffeomorphism $f:\partial C\to\partial C$ which does not extend to a diffeomorphism of $C$. No finite-order condition on $f$, and no Stein structure on $C$, is imposed. Contractible means homotopy equivalent to a point.

Given a smooth embedding $e:C\hookrightarrow W$ into the interior of a closed smooth $4$-manifold, define the cork twist \[ W_{e,f}=(W\setminus\operatorname{int}(e(C)))\cup_{e|_{\partial C}\circ f}C. \] The boundary identification replaces the original attachment by the specified diffeomorphism $f$; smooth collars are used for the gluing.

Does there exist one cork $(C,f)$ with the property that for every pair $W,W'$ of closed simply connected smooth $4$-manifolds which are homeomorphic but not diffeomorphic, some embedding $e:C\hookrightarrow W$ satisfies $W_{e,f}\cong W'$ by a diffeomorphism? The cork and boundary map must be fixed before the pair is chosen. Only their embedding in $W$ may vary with the pair.

Universality asks for a single twist to realize each requested change, not a sequence of different corks or arbitrary powers of $f$. The question concerns closed ambient manifolds; a version requiring universality for manifolds with boundary is a different assertion. No fixed homeomorphism between $W$ and $W'$ is required to be realized by the surgery.

Short English statement

Is there one fixed contractible cork and boundary twist capable of relating every exotic pair of closed simply connected four-manifolds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2890 (KP-4.14), 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 4.14. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.