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

---


---

- Tags 
	- #homotopy/bundles 
- Refs:
	- #todo/add-references
- Links:
	- [K-theory](Unsorted/K-theory.md)
	- [characteristic class](Unsorted/characteristic%20class.md)
	- [vector bundle](Unsorted/vector%20bundles.md)
	- [Thom space](Unsorted/Thom%20space.md)

---

# Chern class

Top Chern class is the [Euler class](Unsorted/Euler%20class.md).

![](attachments/Pasted%20image%2020220403212302.png)

![](attachments/Pasted%20image%2020220913162513.png)

![](attachments/Pasted%20image%2020220403210944.png)
![](attachments/Pasted%20image%2020220403211003.png)


![](attachments/Pasted%20image%2020210613125145.png)

![](attachments/Pasted%20image%2020210613125237.png)

# Properties

![](attachments/Pasted%20image%2020210613125300.png)
![](attachments/Pasted%20image%2020210613125002.png)

# Description in terms of $\BU_n$

![](attachments/Pasted%20image%2020220403211515.png)
![](attachments/Pasted%20image%2020220403211527.png)

# Stability

![](attachments/Pasted%20image%2020220403211042.png)

# Splitting principle

![](attachments/Pasted%20image%2020220403211858.png)
![](attachments/Pasted%20image%2020220403211911.png)
 See [flag variety](flag%20variety).

# Formulas

 ![](attachments/Pasted%20image%2020220403212752.png)
![](attachments/Pasted%20image%2020220403213038.png)

# Examples


![](attachments/Pasted%20image%2020210613124925.png)

![](attachments/Pasted%20image%2020210613124952.png)

#examples/explicit-computations 

![](attachments/Pasted%20image%2020220403211426.png)


# Exercises

#examples/exercises 

![](attachments/Pasted%20image%2020210613125327.png)