TITLE: An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space
QUESTION [17 upvotes]: 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.

REPLY [7 votes]: The Dold manifold $P(1,2)$ is a simple example. It is defined by quotienting the ${\rm Spin}^c$ 5-manifold $S^1\times \mathbb{CP}^2$ via the (fixed points free and orientation preserving) involution $(x,z) \mapsto (-x,\bar z)$. In general this construction works for every $P(m,n)=S^m \times \mathbb{CP}^n$ with $m$ odd and $n$ even.
See this paper of Plymen from 1986, Section 4.7