---
tag: 8H58Q6YJCH
kind: concept
name: 4-manifold
msc: [57K40]
specializes: [QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/4-manifold
same_as:
  wikidata: Q2566544
  wikipedia: 4-manifold
---

## Summary

A $4$-manifold is a [[topological manifold|QQQ9B1DK4P]] of dimension $4$.
Dimension $4$ is special: the methods that classify manifolds of dimension at least $5$ fail, and topological and smooth $4$-manifolds behave very differently.
Freedman classified simply connected closed topological $4$-manifolds through their [[intersection form|WK8GKRKAVD]], together with the Kirby–Siebenmann invariant when the form is odd.
[[Donaldson's theorem|34B7K77P31]] and Seiberg–Witten theory show that many of them have no smooth structure, or many smooth structures.
Even $\R^4$ has [[exotic smooth structures|X44PNC38E2]], which happens in no other dimension.

## Explanations

### L1 human

A $4$-manifold is a space that looks like $4$-dimensional space near each point.
Examples are the $4$-sphere, the product $S^2 \times S^2$ of two spheres, and the complex projective plane.
Dimension $4$ is the hardest one in topology.
There is too little room to untangle things as in higher dimensions, and too much room for the tools of dimensions $2$ and $3$.
Ideas from physics, such as gauge theory, gave the main tools for smooth $4$-manifolds.