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Results tagged with spin-geometry
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user 318
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.
16
votes
Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions
The condition for having a $Pin^+$-structure is the vanishing of $w_2$, and for having a $Pin^-$-structure is the vanishing of $w_2 +w_1^2$, for manifolds of any dimension. This is because the Lie gro …
- 18.2k
12
votes
Accepted
Spin structures and quadratic forms on surfaces
It is not quite true. There is an algebraic gadget one can produce from a Spin structure on a surface with boundary (where we fix the Spin structure along the boundary), and one gets a bijection from …
- 18.2k
3
votes
Spin structure on mapping torus
You can do this iff the spin structures $\mathfrak{s}$ and $f^*(\mathfrak{s})$ are isomorphic.
When $X$ is the 2-torus the set of Spin structures is naturally in bijection with $\mathbb{Z}/2 \oplus \ …
- 18.2k
2
votes
Injectivity of the $\alpha$-genus
No. See for example the work of Anderson, Brown, and Peterson
https://projecteuclid.org/euclid.bams/1183527786
- 18.2k