# Contents

## Statement

The Freudenthal suspension theorem (Hans Freudenthal, 1937) is the following theorem about homotopy groups of spheres:

The suspension homomorphism $\sigma :\pi_{n+k}(S^n)\to \pi_{n+k+l}(S^{n+l})$ is an isomorphism for $n\gt k+1$.

In this statement, one can replace $S^n$ with any $(n-1)$-connected space $Y$ while replacing $S^{n+l}$ with the corresponding suspension $\Sigma^{l} Y$.

This theorem justifies introducing the stable homotopy groups of spheres $\pi_k(S):=\pi_{n+k}(S^n)$, as well as stable homotopy groups $\pi_k^S(Y) = \pi_{n+k}(\Sigma^n Y)$, both independent of $n$ where $n\gt k+1$.

## Relation to the Blakers-Massey theorem

The Frudenthal suspension theorem is a special case of the Blakers-Massey theorem.

## References

A formalization in homotopy type theory in Agda is in

Revised on September 29, 2013 16:23:33 by Urs Schreiber (89.204.130.128)