---
tag: X44PNC38E2
kind: object
name: Exotic R4
msc: [57R55, 57K40]
instance_of: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Exotic_R4
same_as:
  wikidata: Q2436606
  wikipedia: Exotic R4
---

## Description

An *exotic $\R^4$* is a [[smooth manifold|F45VQ47GPV]] that is [[homeomorphic|0QT00PB5G2]] to the [[Euclidean space|0XPR1E85W0]] $\R^4$ but not [[diffeomorphic|0KZZ70ZHXP]] to $\R^4$ with its standard smooth structure.
Exotic $\R^4$'s exist.
They were found in the early 1980s as a consequence of [[Freedman's classification|HSD6FKCYRC]] of topological $4$-manifolds and [[Donaldson's theorem|34B7K77P31]] on smooth ones.

## Remarks

For $n \ne 4$, every smooth manifold homeomorphic to $\R^n$ is diffeomorphic to $\R^n$, so dimension $4$ is the only exception.
Clifford Taubes showed that there are uncountably many pairwise non-diffeomorphic exotic $\R^4$'s.
Some of them, the small exotic $\R^4$'s, embed smoothly as open subsets of the standard $\R^4$; others, the large ones, do not.
The constructions use handle decompositions and Casson handles rather than explicit coordinates.

## Explanations

### L1 human

Four-dimensional space can be given smooth structures, ways of doing calculus, that are different from the usual one.
Each of them is the same as ordinary $\R^4$ for continuous deformation, but no smooth change of coordinates turns it into ordinary $\R^4$.
This happens in no other dimension, and it is one of the strangest facts of geometry.