Smooth 4-dimensional Poincare conjecture

Definitions and mathematical statement

Let $S^4=\{x\in\mathbb R^5:\|x\|=1\}$ with its standard smooth structure. A smooth homotopy four-sphere is a closed, connected, smooth four-dimensional manifold $M$ homotopy equivalent to $S^4$. Closed means compact without boundary. Homotopy equivalence means there are continuous maps $f:M\to S^4$ and $g:S^4\to M$ with $g\circ f$ and $f\circ g$ homotopic to the respective identities.

The smooth four-dimensional Poincaré conjecture is \[ M\simeq S^4\quad\Longrightarrow\quad M\cong_{\mathrm{diff}}S^4 \] for every such $M$. A diffeomorphism is a bijective smooth map with smooth inverse. No particular homotopy equivalence is required to become the diffeomorphism, and no orientation is prescribed.

The smooth category and dimension four are part of the target. A homeomorphism alone does not identify differentiable structures. The statement also cannot be tested using noncompact manifolds or manifolds with boundary, since neither is in the domain of the quantifier. A negative answer must supply a closed smooth homotopy four-sphere and an obstruction to every diffeomorphism to the standard sphere. Merely a nonstandard presentation, handle picture, or triangulation does not establish such an obstruction.

Short English statement

Is every closed smooth four-manifold with the homotopy type of the four-sphere diffeomorphic to the standard four-sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3411 (OPG-37125), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Smooth 4-dimensional Poincare conjecture. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/smooth_4_dimensional_poincare_conjecture.