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Michael Albanese
Michael Albanese
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A An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space

Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?

Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not spin but it is double covered by a K3 surface which is spin.

Every orientable manifold of dimension at most four is spin${}^c$, so any such examples must have dimension at least five. The only non-spin${}^c$ manifold I know of is the Wu manifold $SU(3)/SO(3)$. Of course taking products or connected sums with other manifolds provides more examples.

A non-spin${}^c$ manifold with a spin${}^c$ covering space

Is there a closed smooth manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?

Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not spin but it is double covered by a K3 surface which is spin.

Every orientable manifold of dimension at most four is spin${}^c$, so any such examples must have dimension at least five. The only non-spin${}^c$ manifold I know of is the Wu manifold $SU(3)/SO(3)$. Of course taking products or connected sums with other manifolds provides more examples.

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

Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?

Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not spin but it is double covered by a K3 surface which is spin.

Every orientable manifold of dimension at most four is spin${}^c$, so any such examples must have dimension at least five. The only non-spin${}^c$ manifold I know of is the Wu manifold $SU(3)/SO(3)$. Of course taking products or connected sums with other manifolds provides more examples.

Source Link
Michael Albanese
Michael Albanese
  • 19.3k
  • 9
  • 87
  • 160

A non-spin${}^c$ manifold with a spin${}^c$ covering space

Is there a closed smooth manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?

Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not spin but it is double covered by a K3 surface which is spin.

Every orientable manifold of dimension at most four is spin${}^c$, so any such examples must have dimension at least five. The only non-spin${}^c$ manifold I know of is the Wu manifold $SU(3)/SO(3)$. Of course taking products or connected sums with other manifolds provides more examples.