---
date: 2022-02-23 18:45
modification date: Friday 25th March 2022 22:14:38
title: "Thom space"
aliases: [Thom space, Thom spectrum, Thom construction, disc bundle, sphere bundle, Thom class, Thom spectra, Thom isomorphism]

---

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


---

- Tags 
	- #homotopy/bundles 
- Refs:
	- #todo/add-references
- Links:
	- [Morava K theory](Unsorted/Morava%20K%20theory.md)
	- [Thom spectra](Unsorted/Pontrayagin-Thom.md)
	- [MO](Unsorted/Pontrayagin-Thom.md)
	- [homotopy equivalent manifolds are cobordant](homotopy%20equivalent%20manifolds%20are%20cobordant.md)
	- [Stiefel-Whitney](Unsorted/Stiefel-Whitney%20class.md)

---

# Thom space

![](attachments/Pasted%20image%2020220410211927.png)

![](attachments/Pasted%20image%2020220403212336.png)
![](attachments/Pasted%20image%2020220403212410.png)

Formed by by collapsing the complement of the normal bundle?

Thought of as a twisted suspension, since for a trivial bundle $B\cross \RR^n  \mapsvia{p} B$ we have $\Th(p) = \Suspend^n B_+$.

Coboridsm classes $\Omega_*$ as stable homotopy groups of $\MO$:

![](attachments/Pasted%20image%2020220508194534.png)

# The Pontrayagin-Thom construction

![](attachments/Pasted%20image%2020220508194647.png)

# Thom spectra

![](attachments/Pasted%20image%2020220508172627.png)

![](attachments/Pasted%20image%2020210511202422.png)

Can also construct $\Th(p)$ by applying a fiberwise one-point compactification on $E$ and identifying all the added points to a single basepoint.

![](attachments/Pasted%20image%2020220325221433.png)

Relation to [topological K theory](Unsorted/topological%20K%20theory.md):
![](attachments/Pasted%20image%2020220325221544.png)

![](attachments/Pasted%20image%2020220325230053.png)

![](attachments/Pasted%20image%2020220404092607.png)

# Orientations

![](attachments/Pasted%20image%2020220325230009.png)

See [complex oriented cohomology theory](Unsorted/complex%20oriented%20cohomology%20theory.md).

Can view as a twisted suspension spectrum?

![](attachments/Pasted%20image%2020220209190728.png)

![](attachments/Pasted%20image%2020220209190807.png)

# The Thom Diagonal

![](attachments/Pasted%20image%2020220325231846.png)
![](attachments/Pasted%20image%2020220325231926.png)

# Product formula

Relates to [smash product](Unsorted/smash%20product.md):
![](attachments/Pasted%20image%2020220403212454.png)

# Thom isomorphism theorem
![](attachments/Pasted%20image%2020220403212524.png)

# Relation to Euler class
![](attachments/Pasted%20image%2020220403212606.png)

# Cofiber sequence
![](attachments/Pasted%20image%2020220403212635.png)

# Examples

![](attachments/Pasted%20image%2020220403212422.png)



![](attachments/Pasted%20image%2020220508194744.png)
![](attachments/Pasted%20image%2020220508194758.png)

![](attachments/Pasted%20image%2020220508194916.png)