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The smooth four-dimensional Poincaré conjecture
Disprove the smooth 4D Poincaré conjecture with two knots sharing a 0-surgery, one slice and one with a nonzero Rasmussen-type invariant.
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| 1 | +# The smooth four-dimensional Poincaré conjecture | |
| 2 | + | |
| 3 | +Is every smooth 4-manifold that is homotopy equivalent to the 4-sphere actually diffeomorphic to it? This smooth four-dimensional Poincaré conjecture is the most important remaining case of the Poincaré conjectures, and most experts believe it is false. Manolescu and Piccirillo proposed a concrete route to a counterexample through pairs of knots; this problem asks to complete it. | |
| 4 | + | |
| 5 | +## Problem | |
| 6 | + | |
| 7 | +**Conjecture (SPC4).** Every smooth closed 4-manifold homotopy equivalent to $S^4$ is diffeomorphic to $S^4$. | |
| 8 | + | |
| 9 | +A knot $K \subset S^3$ is **slice** if it bounds a smoothly embedded disk in $B^4$. | |
| 10 | + | |
| 11 | +**Task.** Disprove SPC4 by exhibiting knots $K_1, K_2 \subset S^3$ such that | |
| 12 | + | |
| 13 | +1. the 0-framed surgeries on $K_1$ and $K_2$ are related by an explicit sequence of Kirby moves; | |
| 14 | +2. $K_1$ is slice, shown by an explicit slice-disk movie (births, band moves and Reidemeister moves ending in an unlink); | |
| 15 | +3. $K_2$ is not slice, shown by a nonzero Rasmussen-type invariant computed from Khovanov homology. | |
| 16 | + | |
| 17 | +If SPC4 held, (1) and (2) would force $K_2$ to be slice, contradicting (3). | |
| 18 | + | |
| 19 | +## Background | |
| 20 | + | |
| 21 | +Topologically, every homotopy 4-sphere is a 4-sphere (Freedman). The smooth version of the Poincaré conjecture is known to hold in dimensions 1, 2, 3, 5, 6, 12, 56 and 61 and to fail in many others, while dimension four remains open (see the Lean file below). Freedman, Gompf, Morrison and Walker examined computer-assisted approaches to SPC4 [2]. Rasmussen's $s$-invariant, extracted from Khovanov homology, vanishes on slice knots [3], so it can certify that a knot is not slice. Manolescu and Piccirillo gave a general construction of pairs of knots with the same zero-surgery and found five topologically slice knots such that, if any one of them were smoothly slice, an exotic 4-sphere would follow [1]. | |
| 22 | + | |
| 23 | +## What is known | |
| 24 | + | |
| 25 | +- SPC4 is open; formal-conjectures states it in Lean as `poincare_conjecture.variants.smooth_dimension_four` in [Poincare.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Millennium/Poincare.lean). | |
| 26 | +- The Manolescu–Piccirillo strategy [1]. Its five original candidates are now excluded: Nakamura proved that none of them is slice, by a method that covers the whole infinite family they come from, while noting that the general strategy is not ruled out ([arXiv:2203.14270](https://arxiv.org/abs/2203.14270), 2022). | |
| 27 | +- Computer searches for ribbon disks by Gukov, Halverson, Manolescu and Ruehle ruled out many potential counterexamples [4]. | |
| 28 | +- Candidate exotic spheres keep turning out standard: in 2026 Oliveira-Smith showed that a homotopy 4-sphere constructed by Dunfield and Gong is diffeomorphic to $S^4$ ([arXiv:2603.23717](https://arxiv.org/abs/2603.23717), preprint). | |
| 29 | +- Epoch notes two risks: no counterexample may exist along this route, and showing that a complicated knot is not slice may be too costly to compute. | |
| 30 | + | |
| 31 | +## What would count as progress | |
| 32 | + | |
| 33 | +- A certificate with the three parts above: the Kirby-move sequence from the 0-framed $K_1$ to the 0-framed $K_2$, the slice movie for $K_1$, and a diagram of $K_2$ with a nonzero invariant. | |
| 34 | +- Epoch scores submissions with its own verifier: it replays every move exactly, checks that the slice surface is a disk (births minus bands plus final components equals 1), and runs KnotJob on $K_2$ within 30 minutes. | |
| 35 | +- New candidate pairs of knots with the same 0-surgery that Nakamura's method cannot exclude. | |
| 36 | + | |
| 37 | +## References | |
| 38 | + | |
| 39 | +1. C. Manolescu and L. Piccirillo, From zero surgeries to candidates for exotic definite four-manifolds, *J. London Math. Soc.* 108 (2023), 2001–2036. [doi:10.1112/jlms.12800](https://doi.org/10.1112/jlms.12800) ([arXiv:2102.04391](https://arxiv.org/abs/2102.04391)) | |
| 40 | +2. M. H. Freedman, R. E. Gompf, S. Morrison and K. Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, *Quantum Topol.* 1 (2010), 171–208. [doi:10.4171/QT/5](https://doi.org/10.4171/QT/5) | |
| 41 | +3. J. Rasmussen, Khovanov homology and the slice genus, *Invent. Math.* 182 (2010), 419–447. [doi:10.1007/s00222-010-0275-6](https://doi.org/10.1007/s00222-010-0275-6) | |
| 42 | +4. S. Gukov, J. Halverson, C. Manolescu and F. Ruehle, Searching for ribbons with machine learning, 2023. [arXiv:2304.09304](https://arxiv.org/abs/2304.09304) | |
| 43 | + | |
| 44 | +## Source | |
| 45 | + | |
| 46 | +Imported from [FrontierMath: Open Problems, smooth-poincare-4d](https://epoch.ai/frontiermath/open-problems/smooth-poincare-4d), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion. | |