TITLE: Spin^c structures on manifolds with almost complex structure
QUESTION [13 upvotes]: 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 $M$ ?

REPLY [8 votes]: 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.