Minimal Exotic Smooth Structures on Closed Four-Manifolds

Definitions and mathematical statement

A closed compact orientable smooth four-manifold is a compact orientable smooth manifold of dimension four with empty boundary. The supplied statement does not specify connectedness. Say that an underlying topological four-manifold admits multiple smooth structures if it supports two smooth structures whose resulting smooth manifolds are homeomorphic but not diffeomorphic. Here diffeomorphisms are not required to preserve an orientation; the source does not specify an orientation-sensitive equivalence relation.

For an orientation of such a manifold $M$, its real intersection form is \[ Q_M\colon H^2(M;{\mathbb{R}})\times H^2(M;{\mathbb{R}})\longrightarrow{\mathbb{R}}, \qquad Q_M(a,b)=\langle a\smile b,[M]\rangle. \] Call $M$ definite if $Q_M$ is positive definite or negative definite. Reversing all component orientations changes the sign of this form. The source does not specify whether manifolds with $H^2(M;{\mathbb{R}})=0$ are included; under the usual universal-inequality definition, definiteness on the zero vector space holds vacuously.

The supplied problem asks for the following.

1. Determine the simplest closed compact orientable smooth four-manifold admitting multiple smooth structures. No complexity invariant, ordering, or equivalence criterion for "simplest" is supplied, so this part does not yet specify a precise minimization problem.
2. Determine whether there exist closed compact orientable smooth four-manifolds $M_0,M_1$ such that \[ M_0\cong_{\mathrm{homeo}}M_1, \qquad M_0\not\cong_{\mathrm{diff}}M_1, \qquad Q_{M_0}\text{ is definite}. \] Orientations may be transported by the homeomorphism when comparing the forms. No restriction on the fundamental group is imposed in the supplied question.

The supplied dated curation labels the problem partially solved and reports examples of closed definite four-manifolds with multiple smooth structures and fundamental group ${\mathbb{Z}}/2{\mathbb{Z}}$. It also reports that the simplest example and the simply connected definite case remain unresolved. These are supplied literature assessments, not independently verified conclusions. The simply connected case is a refinement mentioned in that assessment, rather than a hypothesis of the original existence question.

Short English statement

The problem asks which closed orientable four-manifold is simplest among those supporting inequivalent smooth structures, and whether such structures occur on a definite four-manifold. The meaning of "simplest" is unspecified. Supplied dated curation reports definite examples with fundamental group ${\mathbb{Z}}/2{\mathbb{Z}}$, while describing the simplest example and the simply connected definite case as unresolved; this assessment requires independent verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004410$, title "Minimal Exotic Smooth Structures on Closed Four-Manifolds," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2020 and does not identify an individual proposer. It labels the statement "exact" and the status partially_solved; these are supplied registry labels.
• Supplied report citation: Low-dimensional Topology (2021), Oberwolfach Reports, pp. 465--516, DOI 10.4171/owr/2020/8. https://doi.org/10.4171/owr/2020/8. The background describes the report as documenting workshop discussions and open problems.
• Supplied literature source: Stipsicz and Szabó, "exotic definite four-manifolds." https://arxiv.org/abs/2310.16156. This is the supplied reference label; a full bibliographic citation is not provided.
• Supplied literature source: "Oberwolfach Report 8/2020." https://doi.org/10.4171/owr/2020/8.
• Literature assessment checked at 2026-08-22: the supplied curation reports definite examples with fundamental group ${\mathbb{Z}}/2{\mathbb{Z}}$, and describes the simplest example and the simply connected definite case as unresolved. This date and assessment are supplied dated curation, not independently verified. The background expressly requests independent verification of time-sensitive status and cited claims.