# Questions tagged [spin-geometry]

For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}^{\pm}(n)$ 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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### Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$

**5**

**1**answer

### Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$

**3**

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### Why in $S^2$ is there no spin structure? [closed]

**6**

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### Spin structure using flag manifolds instead of a Riemannian metric

**9**

**1**answer

### Topological Spin manifolds in dimension 4

**4**

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### Understanding $w_2$ as an obstruction to trivializing the tangent bundle over 2-cells

**6**

**1**answer

### Spectral gaps for spin manifold Laplace spectrum

**6**

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### Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?

**3**

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### Embedding of Riemannian symmetric spaces $E_I$ and $E_{IV}$ into Lie group $E_6$

**2**

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### First Chern Class of Contact Structure which is not Torsion

**9**

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### An equivalent definition for $\text{Spin}^c$-structures

**2**

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### Existence of a certain kind of compact spin manifold with boundary

**6**

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### Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$

**10**

**1**answer

### Discrete Pin structures

**4**

**1**answer

### Spin groups in terms of matrices and/or linear operators

**5**

**0**answers

### Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

**6**

**2**answers

### Lifting a diffeomorphism into a spinor bundle automorphism

**2**

**0**answers

### Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

**2**

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### The complex Clifford algebra

**3**

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### Pairing the Arf with Stiefel-Whitney class

**2**

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### A generalization of the Clifford algebra

**3**

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### The Dirac-Ricci operator

**4**

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

**2**

**0**answers

### Is there an analog of a Chern-Simons formula for the pfaffian $Pf(F)$ of a $SO(2n)$ curvature $F$?

**12**

**1**answer

### Obstruction of spin-c structure and the generalized Wu manifods

**9**

**3**answers

### Spin-H structures

**8**

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### Are spin Hurwitz numbers $r$-spin Hurwitz numbers?

**1**

**1**answer

### Example of a certain partitioned manifold

**7**

**1**answer

### What is a formal definition of a Fermionic quantum field?

**3**

**2**answers

### $spin_{\mathbb{C}}$ Connection and Charge Parity

**7**

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### Vanishing of K-theoretic index and positive scalar curvature

**6**

**1**answer

### Lickorish-Wallace theorem for torsion spin$^c$ 3-manifolds?

**3**

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### Noncompact dual of $\mathrm{Spin}(2n)$ corresponding to $\mathfrak{so}^*(2n)$

**3**

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### Causal fermion systems fromm fractal geometry

**6**

**1**answer

### Spinor bundle tensored with certain line bundle gives the dual spinor bundle

**7**

**1**answer

### First Chern class of a specific line bundle

**6**

**2**answers

### Action of the spin covariant derivative on gamma matrices?

**16**

**1**answer

### An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space

**3**

**1**answer

### Injectivity of the $\alpha$-genus

**8**

**0**answers

### Dixmier-Douady class is the third integral Stiefel-Whitney class

**3**

**1**answer

### Spin Structure on AdS- Schwarzschild manifold

**6**

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### Has the structure of the 2-dimensional pin$^{\pm}$ bordism categories been written down?

**4**

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### Section of the spinor bundle over $S^{1}$ that extend to sections of the spinor bundle over $D^{2}$

**8**

**1**answer

### K-homology classes of Dirac operators on Hermitian manifolds

**8**

**0**answers

### Regularilty of Commutative Spectral Triples

**8**

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### Sections of “forgetful” projections between flag manifolds

**2**

**2**answers

### Matrix expression for elements of $\text{SO}_0(1,4)$

**9**

**1**answer

### Is a 4-dimensional submanifold of a spin manifold always spin?

**12**

**1**answer

### Spin structures on Sasakian manifolds and the Kähler analogy

**2**

**0**answers