---
tag: 6WYEMXSHD9
kind: object
name: Exotic sphere
msc: [57R60, 57R55]
instance_of: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Exotic_sphere
same_as:
  wikidata: Q3105146
  wikipedia: Exotic sphere
---

## Description

An *exotic sphere* is a [[smooth manifold|F45VQ47GPV]] $\Sigma$ that is [[homeomorphic|0QT00PB5G2]] to the standard sphere $S^n$ but not [[diffeomorphic|0KZZ70ZHXP]] to it.
Equivalently, it is $S^n$ with a [[smooth structure|R2TQ82PCNB]] that is not diffeomorphic to the standard one.

Milnor found the first examples in dimension $7$.
He took total spaces of bundles over $S^4$ with fibre $S^3$.
For suitable bundles he built a Morse function with exactly two critical points, so the total space is homeomorphic to $S^7$.
He then defined an invariant of such a manifold from a compact $8$-manifold that it bounds, using the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].
This invariant is zero for the standard $S^7$ and nonzero for some of the bundles, so these are not diffeomorphic to $S^7$.

## Remarks

Kervaire and Milnor showed that for $n \ge 5$ the oriented diffeomorphism classes of smooth homotopy $n$-spheres form a finite abelian group $\Theta_n$ under [[connected sum|Y0K0K5NGWW]].
For $n = 7$ this group is cyclic of order $28$.
For $n = 1, 2, 3, 5, 6$ there are no exotic spheres.
In dimension $4$ it is unknown whether exotic spheres exist; this is the [[smooth four-dimensional Poincaré conjecture|P70VGYNFS5]].

## Explanations

### L1 human

An exotic sphere is a space that is a sphere for a topologist who allows bending and stretching, but not for one who also demands smoothness.
There is no smooth way to straighten it into the round sphere.
The first one appears in dimension seven.
Up to orientation-preserving diffeomorphism, there are $28$ different smooth versions of the $7$-sphere.