Math 2.0

Manifolds and cell complexes Four-manifolds Conjecture

Smooth four-dimensional Poincaré conjecture

Builds on 4-manifoldSmooth manifoldn-sphereHomotopy equivalenceDiffeomorphismSmooth structure

Statement

Let MM be a closed smooth 4-manifold that is homotopy equivalent to the 44-sphere S4S^4. Then MM is diffeomorphic to S4S^4 with its standard smooth structure.

Remarks

The conjecture is open. Its topological version is true: Michael Freedman proved in 1982 that such an MM is homeomorphic to S4S^4. So the conjecture is equivalent to the statement that S4S^4 has no exotic smooth structure. In dimensions 11, 22, 33, 55 and 66 the smooth version holds, while in dimension 77 it fails, by the exotic spheres of John Milnor. Some proposed counterexamples, among them an infinite family of Cappell–Shaneson homotopy spheres, have been shown to be standard. Gauge-theoretic invariants such as the Seiberg–Witten invariants need b2+≥1b_2^+ \ge 1, so they say nothing about homotopy 44-spheres.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
4-manifold, concept, no evidence level4-manifoldconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptSmooth structure, concept, no evidence levelSmooth structureconceptn-sphere, object, no evidence leveln-sphereobjectTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptDiffeomorphism, concept, no evidence levelDiffeomorphismconceptSmooth four-dimensional Poincaré conjecture, conjecture, no evidence levelSmooth four-dimension…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. 4-manifold concept
  2. Euclidean space object
  3. Function concept
  4. Homotopy equivalence concept
  5. Smooth manifold concept
  6. Smooth structure concept
  7. n-sphere object
  8. Total derivative concept
  9. Function of several variables concept
  10. Differentiable function of several variables concept
  11. Diffeomorphism concept
  12. Smooth four-dimensional Poincaré conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture
About this page
TagP70VGYNFS5
URIm2://P70VGYNFS5
Kindconjecture
WritingW0 stub
MSC57R60, 57R55, 57K40
StatusThis copy of the site has no build data. Run m2 status P70VGYNFS5.
Aboutn-sphere, 4-manifold
Uses4-manifold, Smooth manifold, n-sphere, Homotopy equivalence, Diffeomorphism, Smooth structure
Changed2 November 2026

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