---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: "Euler class"
aliases: [Euler class]

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


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- Tags 
	- #homotopy/bundles 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# Euler class

- The Euler class of the tangent bundle of an oriented [manifold](manifold) can be identified with the [Euler characteristic](Euler%20characteristic) times the [fundamental class](fundamental%20class.md)

- Fact: $\hat\T S^2 \isoas{\Diff} \RP^3$ are diffeomorphic, where the LHS is the unit tangent bundle.
![](attachments/Pasted%20image%2020220403210801.png)

Relation to [BunG](Unsorted/BunG.md):
![](attachments/Pasted%20image%2020220403210833.png)

![](attachments/Pasted%20image%2020220403193727.png)
![](attachments/Pasted%20image%2020220403193822.png)

Relation to the [Gysin sequence](Unsorted/Gysin%20sequence.md):
![](attachments/Pasted%20image%2020220403193844.png)

![](attachments/Pasted%20image%2020220403193904.png)

![](attachments/Pasted%20image%2020220403193926.png)

# Formulas

![](attachments/Pasted%20image%2020220403212713.png)
![](attachments/Pasted%20image%2020220403212719.png)