# 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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### About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold

I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that Indeed, on an orientable 3-manifold, the eigenvalues of the Dirac ...
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### Is there an analog of a Chern-Simons formula for the pfaffian $Pf(F)$ of a $SO(2n)$ curvature $F$?

..something similar to $tr(A \wedge dA + 2/3 * A \wedge A \wedge A)$ for $n = 2$ ?
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### Dixmier-Douady class is the third integral Stiefel-Whitney class

Let $M$ be (say smooth) manifold. From the short exact sequence of groups $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}_2 \to 0$ (where the first map is multiplication by $2$) one obtain long exact ...
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### Is a 4-dimensional submanifold of a spin manifold always spin?

Let $M^d$ be a $d$-dimensional orientable spin manifold, and $N^4$ is a closed $4$-dimensional orientable submanifold of $M^d$. Is $N^4$ always spin? If $d=5$, is $N^4$ always spin? If $N^4$ is a ...
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A Sasakian manifold is often said to be the odd dimensional analogue of a Kähler manifold. Now for a $2n$-dimensional Kähler manifold we know from Atiyah that it is spin exactly if the line bundle $\... 0answers 51 views ### Understanding the relationship between Spin$^c$orientations and Spin$^c$structures I'm looking for some guidance in understanding and writing down a proof of the following statement, concerning the relationship between Spin$^c$structures and Spin$^c$orientations, from an ... 1answer 288 views ### Lagrangian Grassmannian as a Spin Manifold I am trying to better understand this nice answer to a question of mine, which states Spin structures on a compact complex manifold$(M^{2n},J)$are in bijective correspondence with isomorphism ... 1answer 570 views ### Which Kahler Manifolds Are Spin? As is well-known (see here for a M.O. question) all Kahler manifolds are$spin^c$. I would like to ask which are in fact$spin$. Taking my motivation from the case of complex projective space, I ... 1answer 71 views ### Trace of the chiral matrix of a subspace Let$(V,Q)$be a pair consisting of a$\mathbb{C}$-vector space$V$together with a nondegenerate bilinear form$Q$and let$V_0\subseteq V$be a linear subspace such that$Q\vert_{V_0}$is ... 2answers 570 views ### Does Spin cobordism vanish in dimension$4k-1$? For the purposes of a remark in a paper in preparation, I would like to know if anyone can confirm that$\Omega^{spin}_{4k-1} = 0$. In the Atiyah-Patodi-Singer paper, Spectral asymmetry and ... 1answer 212 views ### Recovering K-theory and KO-theory from KR-theory and Bott Periodicity Theory Reference: H. Blaine Lawson, Spin Geometry, Page 72 Problem: Here Remark 10.5 states an internal symmetry in the KR-theory that for any compact space$X$there are isomorphisms$$KR(X\times S^{0,p}) \... 1answer 302 views ### Topological obstruction for the existence of spin$^c\$ structure

Recently I asked on stack exchange the following question: https://math.stackexchange.com/questions/2088888/vanishing-of-certain-cohomology-class-and-existence-of-spin-structure I would like to know ...