---
date: 2022-04-05 19:51
modification date: Tuesday 5th April 2022 19:51:21
title: "equivariant homotopy theory"
aliases: [equivariant homotopy theory]

---

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


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- Tags 
	- #todo/untagged
- Refs:
	- Lecture notes: <https://iwoat.github.io/> #resources/notes
- Links:
	- [equivariant cochains](Unsorted/equivariant%20cochains.md)
	- [equivariant homotopy group](equivariant%20homotopy%20group.md)
	- [regular representation](regular%20representation)
	- [sign representation](sign%20representation)
	- [Burnside ring (homotopy theory)](Burnside%20ring%20(homotopy%20theory))
	- [norm](Unsorted/homotopy%20fixed%20points.md)

---

# equivariant homotopy theory


Via CW complexes:
![](attachments/Pasted%20image%2020220425092601.png)
![](attachments/Pasted%20image%2020220425092615.png)

# G-spectra

![](attachments/Pasted%20image%2020220508213101.png)

# Values

![](attachments/Pasted%20image%2020220510184000.png)
![](attachments/Pasted%20image%2020220510184040.png)

# Relation to cohomology

Eilenberg-Maclane equivariant spectra represent [Bredon cohomology](Unsorted/Bredon%20cohomology.md):
![](attachments/Pasted%20image%2020220510191534.png)

![](attachments/Pasted%20image%2020220510191820.png)
![](attachments/Pasted%20image%2020220510191910.png)

# Examples

![](attachments/Pasted%20image%2020220510184124.png)
![](attachments/Pasted%20image%2020220510184134.png)
![](attachments/Pasted%20image%2020220510185229.png)

## C_2

![](attachments/Pasted%20image%2020220510191218.png)
![](attachments/Pasted%20image%2020220510191244.png)
![](attachments/Pasted%20image%2020220510191255.png)

## Eilenberg-Maclane

![](attachments/Pasted%20image%2020220510191441.png)
![](attachments/Pasted%20image%2020220510191424.png)