---
date: 2022-02-23 18:45
modification date: Friday 1st April 2022 21:14:59
title: "Obstruction theory in homotopy"
aliases: [Postnikov tower, Whitehead tower, Postnikov, "k-invariant", "k invariant", "k invariants"]

---

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


---

- Tags 
	- #homotopy 
- Refs:
	- Lecture Notes: <https://webspace.science.uu.nl/~caval101/homepage/Geometry_and_Topology_2015_files/Groth - lecture notes on homotopy theory.pdf>
		- Includes Postnikov and Whitehead towers
	- <https://www.math.purdue.edu/~zhang24/towers.pdf>
	- Obstruction theory notes: <https://stanford.edu/~sfh/282B.pdf#page=35>
- Links:
	- #todo/create-links 

---

# Obstruction Theory

# Motivations

![](attachments/Pasted%20image%2020220422204833.png)

The rough idea of obstruction theory is simple. 
Suppose we want to construct some kind of function on a [CW complex](CW%20complex.md) $X$. We do this by induction: if the function is defined on the k-skeleton $X_k$, we try to extend it over the $(k + 1)\dash$skeleton $X^{k+1}$. The obstruction to extending over a $(k + 1)\dash$cell is an element of $\pi_k$ of something. These obstructions fit together to give a cellular cochain on $X$ with coefficients in this $π_k$. In fact this cochain is a cocycle, so it defines an “obstruction class” in $H_{k+1}(X; π_k(something))$. If this cohomology class is zero, i.e. if there is a cellular $k\dash$cochain $η$ with $0 = δη$, then $η$ prescribes a way to modify our map over the $k\dash$skeleton so that it can be extended over the $(k + 1)\dash$skeleton

# Postnikov tower


![Pasted image 20210505014637.png](Pasted%20image%2020210505014637.png)

![Pasted image 20210505014732.png](Pasted%20image%2020210505014732.png)

[k-invariant](k-invariant):

![Pasted image 20210505014833.png](Pasted%20image%2020210505014833.png)

# Construction

![](attachments/Pasted%20image%2020220401211457.png)
![](attachments/Pasted%20image%2020220401211842.png)
![](attachments/Pasted%20image%2020220401211910.png)
![](attachments/Pasted%20image%2020220401211928.png)

# Whitehead tower

![](attachments/Pasted%20image%2020220401214606.png)

![](attachments/Pasted%20image%2020220403212954.png)

# Unsorted

![](attachments/Pasted%20image%2020220401220151.png)
![](attachments/Pasted%20image%2020220401220201.png)
![](attachments/Pasted%20image%2020220401220228.png)
![](attachments/Pasted%20image%2020220401220239.png)
![](attachments/Pasted%20image%2020220401220301.png)
![](attachments/Pasted%20image%2020220401220311.png)