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date: 2022-02-23 18:45
modification date: Sunday 3rd April 2022 17:50:51
title: "reduction of structure group"
aliases: [structure group, reduction of structureo]

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- Refs:
	- #todo/add-references
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# reduction of structure group

- For $\xi = (E \mapsvia{p} B)$ a real $n\dash$plane bundle, $E$ is orientable iff $?$
- A principal $\SO_n$ bundle lifts to a principal $\Spin_n$ bundle iff the 2nd Stiefel-Whitney class vanishes, $w_2(P) \in H^2(X; \FF_2)$.
	- Choices of spin structures biject with $H^1(X; \FF_2)$.
![](attachments/Pasted%20image%2020220403175050.png)