---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: "spin"
aliases: [spin, spin manifold, spin structure, spinc, spinc structure]

---

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


---

- Tags 
	- #geomtop 
- Refs:
	- [http://math.mit.edu/~mbehrens/18.906spring10/prin.pdf](http://math.mit.edu/~mbehrens/18.906spring10/prin.pdf): #resources/notes 
- Links:
	- [Seiberg-Witten theory](Seiberg-Witten%20theory)
	- [Pin group](Pin%20group.md)

---

# spin

![](attachments/2021-10-03_14-44-11.png)

![attachments/Pasted image 20210612233405.png](attachments/Pasted%20image%2020210612233405.png)

![attachments/Pasted image 20210613130400.png](attachments/Pasted%20image%2020210613130400.png)

- The questions on existence and classification of spin structures may be completely answered in terms of the Stiefel-Whitney classes.
- Spinnable (admits spin structure) implies admitting a $\Spinc$ structure, but not conversely 
- Every closed oriented 4-manifold admits a $\Spinc$ structure
	- There are closed non-oriented manifolds that are not spinnable
- Every closed oriented is spinnable.

![attachments/Pasted image 20210613130528.png](attachments/Pasted%20image%2020210613130528.png)
![attachments/Pasted image 20210613130534.png](attachments/Pasted%20image%2020210613130534.png)
![attachments/Pasted image 20210613130544.png](attachments/Pasted%20image%2020210613130544.png)

- Explanation of [string structure](string%20structure) by vanishing of [characteristic classes](characteristic%20class.md), using the [Whitehead tower](Unsorted/Obstruction%20theory%20in%20homotopy.md). Essentially it all depends on $\pi_* \Orth_n$:

![Pasted image 20211117172353.png](Pasted%20image%2020211117172353.png)

![](attachments/Pasted%20image%2020220325220459.png)

![](attachments/Pasted%20image%2020220325220606.png)

![](attachments/Pasted%20image%2020220408205812.png)
![](attachments/Pasted%20image%2020220408205818.png)

# Spin^c

![](attachments/Pasted%20image%2020220503132644.png)