---
date: 2022-04-03 18:47
modification date: Sunday 3rd April 2022 18:47:07
title: "examples of spectral sequences"
aliases: [examples of spectral sequences]

---

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


---

- Tags 
	- #homotopy 
- Refs:
	- #todo/add-references
- Links:
	- [spectral sequence](Unsorted/spectral%20sequence.md)
	- [Exact Couples](Archive/0200_Stable%20Homotopy%20Seminar%202021/Exact%20Couples.md)
	- [Atiyah Hirzebruch spectral sequence](Unsorted/Atiyah%20Hirzebruch%20spectral%20sequence.md)
	- [Gysin sequence](Gysin%20sequence)
	- [spectral sequence of a double complex](spectral%20sequence%20of%20a%20double%20complex.md)
	- [examples of cohomology rings](examples%20of%20cohomology%20rings.md)

---

# examples of spectral sequences


[Proving Hurewicz](https://people.math.wisc.edu/~maxim/spseq.pdf#page=6): ![](attachments/Pasted%20image%2020220422205054.png)

# Evenly supported complexes
![](attachments/Pasted%20image%2020220403184846.png)

# Loop spaces of spheres

#examples/explicit-computations

![](attachments/Pasted%20image%2020220403184709.png)
![](attachments/Pasted%20image%2020220403184744.png)
![](attachments/Pasted%20image%2020220403184814.png)

Computing 
$$
H_* \Loop S^n = \gens{ x_{n-1}, x_{2(n-1)},  x_{3(n-1)}, \cdots}
$$
![](attachments/Pasted%20image%2020210613215050.png)

## Multiplicative structure

Appearance of a [divided power algebra](divided%20power%20algebra).

![](attachments/Pasted%20image%2020220403195543.png)

![](attachments/Pasted%20image%2020220403195645.png)

# Computing for [Eilenberg-MacLane spaces](Unsorted/Eilenberg-MacLane%20spaces.md)

#examples/explicit-computations 

Computing 
$$
H_*( K(\ZZ, 2); \ZZ) = \ZZ\adjoin{x}, \qquad \abs{x} = 2
$$
![](attachments/Pasted%20image%2020210613214829.png)

![](attachments/Pasted%20image%2020220403203030.png)

# Group cohomology

Used to study [Serre classes](Unsorted/Serre%20class.md):

![](attachments/Pasted%20image%2020220403200244.png)
![](attachments/Pasted%20image%2020220403200308.png)

# p torsion in $\pi_{2p}S^3$

An example of [working one prime at a time](working%20one%20prime%20at%20a%20time):

![](attachments/Pasted%20image%2020220403205144.png)
![](attachments/Pasted%20image%2020220403205152.png)