---
tag: P70VGYNFS5
kind: conjecture
name: Smooth four-dimensional Poincaré conjecture
msc: [57R60, 57R55, 57K40]
uses: [8H58Q6YJCH, F45VQ47GPV, T62369RFH6, XZG6MTSWCN, 0KZZ70ZHXP, R2TQ82PCNB]
about: [T62369RFH6, 8H58Q6YJCH]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture
history:
  posed_by: "not attributed to one person; it is the smooth case in dimension 4 of the generalized Poincaré conjecture"
---

## Statement

Let $M$ be a closed [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] that is [[homotopy equivalent|XZG6MTSWCN]] to the [[$4$-sphere|T62369RFH6]] $S^4$.
Then $M$ is [[diffeomorphic|0KZZ70ZHXP]] to $S^4$ with its standard [[smooth structure|R2TQ82PCNB]].

## Remarks

The conjecture is open.
Its topological version is true: Michael Freedman proved in 1982 that such an $M$ is homeomorphic to $S^4$.
So the conjecture is equivalent to the statement that $S^4$ has no exotic smooth structure.
In dimensions $1$, $2$, $3$, $5$ and $6$ the smooth version holds, while in dimension $7$ 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|PRDDMSTR4X]] need $b_2^+ \ge 1$, so they say nothing about homotopy $4$-spheres.

## Explanations

### L1 human

A homotopy sphere is a space that cannot be told apart from a sphere by the methods of algebraic topology.
In dimension four, every such space is known to be a sphere up to continuous deformation.
The open question is whether it is always a sphere in the smooth sense, where one also keeps track of calculus.