Timeline for Spin structure using flag manifolds instead of a Riemannian metric
Current License: CC BY-SA 4.0
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| Feb 13, 2020 at 21:07 | comment | added | Pierre PC | @JohannesEbert Ah, true, thank you! And in fact the flag bundle $E$ should be the adjoint bundle associated to the action of $\mathrm{GL}^+(n)$ on the flag manifold of $\mathbb R^n$, when considering the whole oriented frame bundle $P_{\mathrm{GL}^+}$. Just to be sure I understand your claim about the structure group, are you saying that the bundle is spin iff its representation as a Cech cocycle with values in $\mathrm{GL}^+(n)$ lifts to a cocycle with values in $G$? Am I right in thinking that this is the standard argument giving the obstruction class for the existence of a spin structure? | |
| Feb 13, 2020 at 19:23 | comment | added | Johannes Ebert | You can instead formulate the spin condition topologically as follows. The group $GL_n(\mathbb{R})^+$ ( matrices of positive determinant) has a unique twofold cover $G$, and a topological spin structure on a vector bundle is a reduction of the structure group to $G$. This is not of much help for index theory, as the spin representation does not extend to $G$. | |
| Feb 13, 2020 at 16:44 | history | asked | Pierre PC | CC BY-SA 4.0 |