---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: "Freudenthal suspension"
aliases: [Freudenthal suspension theorem, Freudenthal suspension]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #homotopy 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# Freudenthal Suspension Theorem
![](attachments/Pasted%20image%2020220403205623.png)
![](attachments/Pasted%20image%2020220403205640.png)

If $X$ is an $n\dash$ connected CW complex, then there are maps $\pi_i X \to \pi_{i+1} \Sigma X$ which is an isomorphism for $i\leq 2n$ and a surjection for $i=2n+1$.

# Examples
![](attachments/Pasted%20image%2020220403205705.png)