---
date: 2022-01-26 17:04
modification date: Wednesday 26th January 2022 21:28:21
title: group cohomology
aliases: [group cohomology, invariants, coinvariants, induction, coinduction, Tate cohomology, crossed homomorphism, central extension]

---

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


---

- Tags 
	- #todo/untagged 
- Refs:
	- <http://www.mathematik.uni-regensburg.de/loeh/teaching/grouphom_ss19/lecture_notes.pdf#page=150> #projects/lecture-notes 
- Links:
	- [Galois cohomology](Galois%20cohomology)
	- [classifying space](Unsorted/classifying%20space.md)
	- [Cartan-Eilenberg formula](Cartan-Eilenberg%20formula)

---

# group cohomology

# Definitions
![](attachments/Pasted%20image%2020220407202723.png)

If $X$ is an [aspherical space](Unsorted/aspherical%20space.md), then $H^k_\sing(X;\mcf) \cong H^k_\Grp(\pi_1 X; \mcf_x)$ where $\mcf$ is a local system.

# Application

See [character variety](Unsorted/character%20variety.md):
![](attachments/Pasted%20image%2020220407200244.png)

# Interpretations

![](attachments/Pasted%20image%2020220403201955.png)
![](attachments/Pasted%20image%2020220403202027.png)

An example of a second group cohomology group is the [Brauer group](Brauer%20group.md): it is the cohomology of the [absolute Galois group](absolute%20Galois%20group.md) of a field $k$ which acts on the invertible elements in a [separable closure](separable%20closure):
\[
H^{2}\left(\operatorname{Gal}(k),\left(k^{\text {sep }}\right)^{\times}\right) .
\]

# Invariants and Coinvariants
![](attachments/Pasted%20image%2020220126170425.png)
![](attachments/Pasted%20image%2020220126212813.png)

# Nonabelian

![](attachments/Pasted%20image%2020220317191230.png)
![](attachments/Pasted%20image%2020220317191345.png)
![](attachments/Pasted%20image%2020220317191407.png)

# Induction and Coinduction

![](attachments/Pasted%20image%2020220126213000.png)
# group cohomology

![](attachments/Pasted%20image%2020220126170447.png)
![](attachments/Pasted%20image%2020220126170511.png)
![](attachments/Pasted%20image%2020220126170630.png)
![](attachments/Pasted%20image%2020220126170642.png)

![](attachments/Pasted%20image%2020220126171131.png)
![](attachments/Pasted%20image%2020220126170956.png)
![](attachments/Pasted%20image%2020220126171115.png)
![](attachments/Pasted%20image%2020220126171203.png)



See [induction](induction)

# Crossed homomorphisms

![](attachments/Pasted%20image%2020220408000706.png)

# Tate Cohomology

![](attachments/Pasted%20image%2020220126213058.png)
![](attachments/Pasted%20image%2020220126213115.png)
![](attachments/Pasted%20image%2020220126213412.png)
# Infty Categorical Coinvariants

![](attachments/Pasted%20image%2020210630222855.png)

# The LES

![](attachments/Pasted%20image%2020220403202448.png)

# Transfers

![](attachments/Pasted%20image%2020220403202507.png)
![](attachments/Pasted%20image%2020220408002033.png)

# Examples

## Finite cyclic groups

#examples/explicit-computations 

![](attachments/Pasted%20image%2020220403202256.png)
![](attachments/Pasted%20image%2020220403202305.png)

## Free groups

![](attachments/Pasted%20image%2020220403202344.png)

# Pairings

![](attachments/Pasted%20image%2020220407202759.png)![](attachments/Pasted%20image%2020220407203954.png)


## Universal coefficients
![](attachments/Pasted%20image%2020220408005152.png)
![](attachments/Pasted%20image%2020220408005207.png)