Contents

# Contents

## Statement

###### Lemma

Let $X$ be an n-connected topological space. Then the adjunction unit of the (suspension $\dashv$ loop space)-adjunction

$X \longrightarrow \Omega \Sigma X$

(e.g. Kochman 96, prop. 3.2.2)

The Freudenthal suspension theorem (Freudenthal 37) is the following theorem about homotopy groups of n-spheres:

###### Proposition
$\pi_{n+k}(S^n) \longrightarrow \pi_{n+k+1}(S^{n+1})$

is an isomorphism for $n\gt k+1$.

More generally, for $X$ an n-connected CW-complex, then the suspension homomorphism on homotopy groups

$\pi_k(X) \longrightarrow \pi_{k+1}(\Sigma X)$

is an isomorphism for $k \leq 2n$.

The suspension isomorphism is equivalently given by the map in lemma .

e.g. (Switzer 75, 6.26)

An alternative proof proceeds from the Blakers-Massey theorem (e.g. Kochman 96, p. 70).

The following more general statement is also often referred to as the Freudenthal suspension theorem:

###### Proposition

For $X$ an n-connected CW-complex and $Y$ a CW-complex of dimension $\leq 2n$, then the maps of homotopy classes of continuous functions

$[Y,X]\stackrel{}{\longrightarrow} [\Sigma Y, \Sigma X] \stackrel{}{\longrightarrow}[\Sigma^2 Y, \Sigma^2 X]$

are isomorphisms. In particular $[Y,X]$ canonically has the structure of an abelian group.

## Properties

### As motivation for stable homotopy theory

The Freudenthal suspension theorem motivated introducing the stable homotopy groups of spheres $\pi_k(S):=\pi_{n+k}(S^n)$, more generally the stable homotopy groups $\pi_k^S(Y) = \pi_{n+k}(\Sigma^n Y)$, both independent of $n$ where $n\gt k+1$, and still more generally the Spanier-Whitehead category, then the stable homotopy category and eventually the stable (infinity,1)-category of spectra.

Due to

Textbook accounts include

A nice expanded version of the latter is in

• Tengren Zhang, Freudenthal suspension theorem (pdf)

A formalization in homotopy type theory in Agda is in

Discussion in equivariant homotopy theory includes