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

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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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# Whitehead's theorem

**Abbreviated statement**: if $X, Y$ are CW complexes, then any map $f: X \to Y$ is a weak homotopy equivalence if and only if it is a homotopy equivalence.

(Note: $f$ must induce maps on all homotopy groups simultaneously.)

**Full Statement**: If $(X, x_0) \mapsvia{f} (Y, f(x_0))$ such that the induced maps
$$
f_*: \pi_*(X, x_0) \to \pi_*(Y, y_0) \\
 [g] \mapsto [f \circ g]
$$
are all isomorphisms and $Y$ is connected, then $f$ is a homotopy equivalence.

![](attachments/Pasted%20image%2020220408201500.png)
![](attachments/Pasted%20image%2020220408201440.png)