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Results tagged with spin-geometry
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user 21564
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.
9
votes
Does Spin cobordism vanish in dimension $4k-1$?
The spin cobordism groups $\Omega^{spin}_n$ have been computed for $n \leq 127$; see section 10 of Secondary Invariants for String Bordism and tmf by Bunke and Naumann. They use MAPLE together with th …
- 19.3k
11
votes
Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions
For $d \geq 4$, let $M_d = (S^1)^{d-4}\times\mathbb{CP}^2$. As tori are parallelisable, $w(M_d) = w(\mathbb{CP}^2)$, in particular $w_1(M_d) = 0$ and $w_2(M_d) \neq 0$, so $M_d$ does not admit a Spin, …
- 19.3k
13
votes
Spin-H structures
As mentioned in Arun Debray's answer, a closed orientable smooth manifold $M$ is spin${}^h$ if and only if there is a principal $SO(3)$-bundle (or equivalently, an orientable real rank three bundle) $ …
- 19.3k
19
votes
Accepted
Is a 4-dimensional submanifold of a spin manifold always spin?
Let $i$ denote an immersion $N \to M$. There is an exact sequence of vector bundles on $N$ given by
$$0 \to TN \to i^*TM \to \nu \to 0$$
where $\nu$ is the normal bundle. As total Stiefel-Whitney c …
- 19.3k
8
votes
Accepted
Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$
A smooth manifold $M$ admits a pin$^+$ structure if and only if $w_2(M) = 0$, and a pin$^-$ structure if and only if $w_1(M)^2 + w_2(M) = 0$; see this page for some information on pin structures. The …
- 19.3k
7
votes
When the Pontryagin square is an even class?
If $B \in H^2(X;\mathbb{Z}_2)$ satisfies $B^2 \neq 0$, then $\mathfrak{P}(B) \in H^4(X; \mathbb{Z}_4)$ is not even as $\mathfrak{P}(B) \equiv B^2 \bmod 2$. Such classes can exist on spin manifolds, fo …
- 19.3k
3
votes
Accepted
Example of a certain partitioned manifold
Assuming that you want $K$ to be a compact manifold with boundary, no such example exists. This is because $\partial K$ is orientedly nullcobordant ($K$ is a cobordism between $\partial K$ and $\empty …
- 19.3k
4
votes
Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for...
Let $\omega = e_1e_2\dots e_{2n-1}e_{2n}$.
For $n > 1$, the center of $Spin(2n)$ is $Z(Spin(2n)) = \{\pm 1, \pm\omega\}$. Note that $\omega^2 = (-1)^n$, so
$$Z(Spin(2n)) = \begin{cases}
\langle -1, \ …
- 19.3k
11
votes
A question about the existence of spin maps
If $M$ and $N$ are spin, then every map between them is a spin map. In particular, there exist spin maps $M \to N$ of degree zero.
If $M$ is spin and $N$ is not spin, then $f : M \to N$ is a spin map …
- 19.3k
5
votes
Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$
The takeaway is that the three classes $w_1$, $w_2$, and $\frac{1}{2}p_1$ are generators for their corresponding cohomology groups. This is the property one needs so that the homotopy fiber is the nex …
- 19.3k