---
date: 2022-02-23 18:45
modification date: Friday 1st April 2022 21:26:14
title: "Hurewicz"
aliases: [Hurewicz theorem, generalized Eilenberg-Maclane spectrum, generalized Eilenberg-Maclane spectra
]

---

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


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- Tags 
	- #homotopy 
- Refs:
	- #todo/add-references
- Links:
	- [Adams resolution](Adams%20resolution.md)

---

# Hurewicz

Given a space $X$, define a family of maps
$$
 h_k: \pi_k X \to H_k X \\
 [f] \mapsto f_*(\mu_k)
$$
where $H_k X = \langle \mu_k \rangle$.

If $X$ is $n-1$ connected where $n\geq 2$, then $h_k$ is an isomorphism for all $k \leq n$.

In particular, $\pi_n X \cong H_n X$ as groups.

![](attachments/Pasted%20image%2020220422205021.png)

Proof using spectral sequences: <https://people.math.wisc.edu/~maxim/spseq.pdf#page=5>

![](attachments/Pasted%20image%2020220401212613.png)
![](attachments/Pasted%20image%2020220401212803.png)
![](attachments/Pasted%20image%2020220401212836.png)

Necessity of simple-connectivity assumption: see the [Poincare homology sphere](Unsorted/Poincare%20homology%20sphere.md)

Relation to the [Whitehead theorem](Unsorted/Whitehead%20theorem.md):
![](attachments/Pasted%20image%2020220403192901.png)
![](attachments/Pasted%20image%2020220403192918.png)


# For spectra

![](attachments/Pasted%20image%2020220508183737.png)
![](attachments/Pasted%20image%2020220508183818.png)