---
date: 2022-02-23 18:45
modification date: Friday 1st April 2022 21:44:53
title: "Eilenberg-MacLane spaces"
aliases: [Eilenberg-MacLane]

---

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


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- Tags 
	- #homotopy 
- Refs:
	- #todo/add-references
- Links:
	- [Moore space](Unsorted/Moore%20spaces.md)
	- [Eckman-Hilton](Eckman-Hilton)
	- [cohomotopy](cohomotopy.md)

---

# Eilenberg-MacLane spaces

![](attachments/Pasted%20image%2020220408200717.png)

# Questions

- Why do Eilenberg-MacLane spaces have complicated higher cohomology?
  - Dually, why do spheres have higher complicated homotopy?

- $\B G \homotopic K(G, 1)$ and $\B^n G \homotopic K(G, n)$ when $n\geq 2$ for $G$ an abelian discrete group.
	- What is $\B^n$?
	- What is $\pi_* \B G$ for $G$ nonabelian and nondiscrete?
	#todo/questions 
	
# Results
- $\Loop^n \B^n G \homotopic G$.

## Uniqueness of E-M Spaces
If $X$ is a space with one nontrivial homology group $G$ in degree $k$, so that $X$ satisfies
$$\pi_i(X) = \cases{G,~i=k\\0,~\text{otherwise}}$$
Then $X \simeq K(G, k)$.

*Note: two spaces with isomorphic homotopy groups may *not* be homotopy-equivalent in general - this is one exception.*

# Construction
 ![](attachments/Pasted%20image%2020220401214411.png)
 ![](attachments/Pasted%20image%2020220401214453.png)
 ![](attachments/Pasted%20image%2020220401214821.png)
![](attachments/Pasted%20image%2020220401214949.png)