# Homotopy of spheres

From Topospaces

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homotopy group and the topological space/family is sphere

Get more specific information about sphere | Get more computations of homotopy group

This article gives the key facts about the homotopy groups of spheres.

## Contents

## Statement

### For

is a two-point set. If we think of as a group, gets the same group structure, namely the structure of the cyclic group of order two. For all , is the trivial group.

### For

is a one-point set (or trivial group, if we choose to use 's group structure to induce a group structure on it). , and is trivial for all .

### For higher

We have that:

- is a one-point set (which we can interpret as a trivial group in some cases).
- is the trivial group for .
- .
- .
- is a finite group for and .

## Proof

### The case

*Fill this in later*

### The case

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### For higher

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