Spin^c structures on manifolds with almost complex structure Let $M$ be a smooth even-dimensional manifold.


*

*Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S_J$ associated to $J$ ? 
(I've read this somewhere but I didn't see the actual construction). 


Is there a way to caracterize that spin$^c$ structure ? (What I mean by that is : is there a theorem of the form : $S_J$ is the unique spin$^c$ structure on $M$ satisfying [a property of compability with $J$]). 


*Suppose $M$ admits an almost-complex structure. Does every spin$^c$ structure comes from an almost-complex structure on $M$ (i.e. is it true that every spin$^c$ structure is equal to a $S_J$ for an almost-complex structure $J$ ?). 
(If not : what happens if we restrict to parallelizable open 4-manifolds ?).

*Same question as 1. 2. but for spin (and not just spin$^c$) structures. EDIT (to clarifiy this question) : Does every spin structure on $M$ comes from has an associated spin^c structure that is associated to an almost-complex structure $J$ on $M$ ?
 A: A $\text{Spin}^c$ structure is equivalent to (a homotopy class of) an almost complex structure on the 2-skeleton of a manifold which extends to the 3-skeleton (except for a surface or when the fiber dimension is odd, where we first need to stabilize the tangent bundle). So in the case of 4-manifolds without 4-handles (in particular 4-manifolds with non-empty boundary) there is a canonical one to one correspondence between $\text{Spin}^c$ structures and almost complex structures. The original reference for this result is Gompf's Spin^c structures and homotopy equivalences (https://doi.org/10.2140/gt.1997.1.41), but there is a more expository version available in Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci and Stipsicz.
A: Assume that $M$ is oriented throughout. Recall that $M$ has a $\text{Spin}^c$ structure iff the third integral Stiefel-Whitney class $\beta w_2 = W_3 \in H^3(M, \mathbb{Z})$ is trivial. Actually more is true: $\text{Spin}^c$ structures on $M$ are in bijection with trivializations of $W_3$, which are a torsor over $H^2(M, \mathbb{Z})$. So we get a functorial way to associate a $\text{Spin}^c$ structure to an almost complex structure for every choice of nullhomotopy of the composite map
$$BU(n) \to BSO(2n) \xrightarrow{W_3} B^3 \mathbb{Z}.$$
Isomorphism classes of nullhomotopies are a torsor over $H^2(BU(n), \mathbb{Z}) \cong \mathbb{Z}$. I don't know in what sense there's a canonical one, but once you pick one you don't have to make any further choices, and in particular you don't have to make any further choices involving $M$. (Edit #2: But see below.)
As for almost complex structures, there is a canonical fiber bundle
$$SO(2n)/U(n) \to BU(n) \to BSO(2n)$$
and almost complex structures on $M$ correspond to sections of the pullback of this bundle along the classifying map $M \to BSO(2n)$ of the tangent bundle. This admits a map of bundles to the corresponding bundle 
$$B^2 \mathbb{Z} \to B \text{Spin}^c(2n) \to BSO(2n)$$
describing $\text{Spin}^c$ structures. The map $SO(2n)/U(n) \to B^2 \mathbb{Z}$ describes a canonical line bundle on the space of linear complex structures on $\mathbb{R}^{2n}$, namely the complex determinant bundle. 
This tells us that we already can't expect all $\text{Spin}^c$ structures to come from almost complex structures when $n = 1$: here $SO(2)/U(1)$ is contractible, so there is in a very strong sense a unique almost complex structure on an oriented surface, but $\text{Spin}^c$ structures are a torsor over $H^2(-, \mathbb{Z})$, which can be nontrivial, e.g. on an oriented closed surface. 
Edit #2: I believe that there is in fact a distinguished nullhomotopy of the composite map $BU(n) \xrightarrow{W_3} B^3 \mathbb{Z}$, as follows. 
We now need the additional assumption that $M$ is Riemannian (although the choice of Riemannian metric won't matter). Then a conceptual description of $W_3 \in H^3(M, \mathbb{Z})$ is that it classifies the bundle of complex Clifford algebras $\text{Cliff}(T_x(M)) \otimes \mathbb{C}$ up to Morita equivalence. Trivializations of this bundle up to Morita equivalence correspond to Clifford module bundles with fiber the unique irreducible representation of $\text{Cliff}(2n) \otimes \mathbb{C}$ (complex spinor bundles). 
If $M$ has an almost complex structure, then a distinguished complex spinor bundle can be constructed out of the complex exterior algebra of the tangent bundle. See Exercise 2.1.37 in Freed's Geometry of Dirac Operators for details. The torsor structure over $H^2(M, \mathbb{Z})$ comes from tensoring this bundle with complex line bundles on $M$, and the action of $H^2(BU(n), \mathbb{Z})$ comes from tensoring this bundle with powers of the canonical bundle. It's possible one might want to do this with a particular power; I'm not sure what's going on here exactly, but see Exercise 2.1.54 in Freed. 
Edit #3: Now that the third question has been clarified, I don't know the answer off the top of my head. In the case of closed oriented surfaces we know that there's a $\mathbb{Z}$'s worth of $\text{Spin}^c$ structures. The unique almost complex structure gives rise to one, and the $2^{2g}$ spin structures also give rise to one (not depending on the spin structure, basically because the Bockstein homomorphism $H^1(M, \mathbb{Z}_2) \to H^2(M, \mathbb{Z})$ is zero in this case). But I don't know if they're the same one; they might differ by a power of the canonical bundle or something (or not, depending on your convention for trivializing $W_3$ as above). In summary, 


*

*Yes, more or less,

*No, and

*I don't know, but I would guess no. 

