# Questions tagged [spin-geometry]

For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}(n)$ and $\operatorname{Spin}^c(n)$. This tag should also be used for any questions about the geometry of spin manifolds, including questions involving Dirac operators and the Lichnerowicz formula.

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### Connection between double cover and spinor fields when there are ramification points?

Apologies, I'm very new and naive to topology. I have been learning about different covering maps between manifolds and recently was learning about the map $T^2 \rightarrow S^2$ which is a two-to-one ...
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### Examples of non-spin 2-manifolds

Do we have some examples of 2d manifolds, which are non-spin (thus $w_2(M) \neq 0$)? thus they must be non-orientable $w_1(M)^2 \neq 0$ thus $w_1(M) \neq 0$. (p.s. every 2d differentiable/triangulable ...
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### Why does the determinant of the Dirac operator on $S^n$ approach $1$ as $n\to\infty$?

Bär and Schopka (reference below) present an interesting conjecture regarding the determinant of the Dirac operator on spheres $S^n$. The conjecture is simple: in the limit $n\to\infty$, the ...
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### Compare two definitions of equivariant index of Dirac operator

Let $(M^{2n},g)$ be a Riemannian manifold with spin structure. Let $G$ is a compact Lie group. Suppose that $G$ acts on $M$ smoothly and the metric is $G$-invariant. Also, assume that the $G$ action ...
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### “enlarged” form and Aut(Spin(8))

In https://en.wikipedia.org/wiki/SO(8)#Spin(8), it says that "Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a ...
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