# Sphere

## Contents

## Definition

### As a subset of Euclidean space

The **unit -sphere** is defined as the subset of Euclidean space comprising those points whose distance from the origin is .

### Inductive definition

Inductively, is defined as a discrete two-point space, and for any natural number , is defined as the suspension of .

This definition is illuminative because many of the results about spheres, particularly those involving algebraic topology and the computation of homology and cohomology, are easily derived from corresponding results about suspensions.

## Particular cases

sphere | |
---|---|

0 | -- discrete two-point space |

1 | circle |

2 | 2-sphere |

3 | 3-sphere |

## Equivalent spaces

Space | How strongly is it equivalent to the circle? |
---|---|

boundary of the -hypercube | homeomorphic; not diffeomorphic because of sharp edges |

boundary of the -simplex | homeomorphic; not diffeomorphic because of sharp edges |

ellipsoid in | equivalent via affine transformation |

one-point compactification of | homeomorphic via stereographic projection |

for : universal cover of real projective space , which is the space of lines in | homeomorphic, diffeomorphic, also isometric if we choose the natural metric. |

## Algebraic topology

### Homology groups

`Further information: homology of spheres`

With coefficients in , the -sphere has and for . In particular, the -sphere is -connected.

Interpretations in terms of various homology theories:

*Fill this in later*

With coefficients in any -module for a ring , the -sphere has and for all .

### Cohomology groups and cohomology ring

`Further information: cohomology of spheres`

With coefficients in , the -sphere has and for . In particular, the -sphere is -connected.

With coefficients in any -module for a ring , the -sphere has and for all .

The cohomology ring is isomorphic to , where is a generator of the cohomology.

### Homotopy groups

`Further information: homotopy of spheres, n-sphere is (n-1)-connected`

For , the homotopy group is the trivial group. , with the identity map being a generator.

The cases are discussed below:

Case | What can we say? |
---|---|

is trivial for all | |

is a finite abelian group |