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Results tagged with spin-geometry
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user 1465
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.
1
vote
Accepted
Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable
For the first question I believe the answer is yes. Almost certainly it's in the literature but I do not know this literature very well.
The idea is as you suggested. Maximal-rank timelike (or space …
- 44.3k
7
votes
Accepted
Is a spin structure on a knot complement the same thing as an orientation of the knot?
Is the theorem true? There is an non-natural bijection. There is no natural bijection.
A link exterior is homotopy-equivalent to a $2$-complex, so a trivialization of the tangent bundle over the $2$ …
- 44.3k
1
vote
Group action on spin^c 4-manifold.
Thanks for clarifying your question.
I rarely think of $spin^c$ structures in terms of principal bundles. A map of a manifold lifts to a map of its principal $SO_n$ bundle if and only if the map i …
- 44.3k
5
votes
Accepted
Spin-c Structures viewed w.r.t. Cell Decomposition
A spin-c structure on a vector bundle $\pi : E \to B$ can be thought of as two things:
(1) A complex line bundle $c : C \to B$
together with
(2) A spin structure on $\pi \oplus c : E \oplus C \to B …
- 44.3k
40
votes
Accepted
What are "good" examples of spin manifolds?
There's the traditional obstruction-theoretic perspective. Orientability means the tangent bundle trivializes over a 1-skeleton. Dually you could think of that as saying the complement of a co-dimen …
4
votes
Accepted
$Spin^c$ structure on the mapping torus of an automorphism of the torus
Yes. Moreover, there's a trivialization of the tangent bundle of $M_\alpha$ that restricts to the standard trivialization of the tangent bundle of $T^2 \times [1/4,3/4]$. By "the standard" trivializ …
- 44.3k
6
votes
Explicit Spin Structures on the Torus
On a torus spin structures have a particularly simple representation, because you have an essentially standard trivialization of the tangent bundle given by your euclidean metric.
From your definiti …
- 44.3k