Open problem, Topology, posed 1961

Smooth four-dimensional Poincaré conjecture

Open

Is every closed smooth $4$ -dimensional manifold $M$ that is homotopy equivalent to the $4$ -sphere $S^4$ (equivalently, by Freedman's theorem, homeomorphic to $S^4$ ) necessarily diffeomorphic to $S^4$ with its standard smooth structure?

Research frontier as of 2026

As of 2026, the smooth four-dimensional Poincaré conjecture remains completely open, with experts divided on whether an exotic $S^4$ exists. On the positive side, Kirby calculus has shown that large classes of potential counterexamples — including Cappell–Shaneson spheres and Gluck twists on ribbon $2$ -knots and spun knots — are standard. On the obstruction side, gauge-theoretic and Floer-homological invariants (Donaldson, Seiberg–Witten, Heegaard Floer) vanish or become trivial on homotopy $4$ -spheres, while Khovanov homology and Rasmussen's $s$ -invariant (introduced to this setting by Freedman, Gompf, Morrison, and Walker in 2010) remain the primary candidate tools for detecting an exotic $S^4$ .

Best known results

Every closed smooth $4$ -manifold homotopy equivalent to $S^4$ is topologically homeomorphic to $S^4$ (Freedman, 1982) and becomes diffeomorphic to $S^4 \# k(S^2 \times S^2)$ after taking a connected sum with finitely many copies of $S^2 \times S^2$ (Wall, 1964).
Gluck twists along all ribbon $2$ -knots, spun knots, and twist-spun knots in $S^4$ , as well as known Cappell–Shaneson families, have been proved diffeomorphic to standard $S^4$ (Gluck 1962, Gordon 1976, Akbulut 2010, Gompf 2010).

Tools and where they stop

Tool	Achieved	Where it stops
Gauge theory and Floer homology (Donaldson, Seiberg–Witten, Ozsváth–Szabó)	Detects infinitely many exotic smooth structures on closed $4$ -manifolds with $b_2^+ > 0$ and uncountably many exotic $\mathbb{R}^4$ s	On a homotopy $4$ -sphere, the second homology group $H_2(M;\mathbb{Z}) = 0$ is trivial, causing Seiberg–Witten and Donaldson invariants to vanish identically
Khovanov skein lasagna modules and Rasmussen's $s$ -invariant (Freedman–Gompf–Morrison–Walker, Manolescu–Neithalath)	Constructs smooth $4$ -manifold invariants and slice-genus bounds directly from combinatorial link diagrams without requiring $b_2^+ > 0$ , distinguishing exotic smooth structures on compact $4$ -manifolds with boundary	Manolescu and Marengon (2020) proved that the original Freedman–Gompf–Morrison–Walker strategy to disprove SPC4 via Rasmussen's $s$ -invariant alone cannot succeed without additional 4-dimensional input

Open questions

Does there exist a smoothly embedded $2$ -sphere $K \subset S^4$ whose Gluck twist $\Sigma_K$ is not diffeomorphic to $S^4$ ?
Is every smoothly embedded $3$ -sphere $S^3 \subset S^4$ smoothly isotopic to the standard equator (the smooth $4$ -dimensional Schoenflies problem)?

References

Herman Gluck (1962). The embedding of two-spheres in the four-sphere · DOI:10.1090/S0002-9947-1962-0146807-0
Michael H. Freedman (1982). The topology of four-dimensional manifolds · DOI:10.2307/429103
Robert E. Gompf, András I. Stipsicz (1999). 4-Manifolds and Kirby Calculus
Michael Freedman, Robert Gompf, Scott Morrison, Kevin Walker (2010). Man and machine thinking about the smooth 4-dimensional Poincaré conjecture · DOI:10.4171/QT/1 · arXiv:0906.5177v2