I am in a certain math conference and we came across Seiberg-Witten equations. Since I am really novice in the field, I asked if all "reasonable" four manifolds carry a $\textit{spin}^{c}$ structure. I was under the impression that $\textit{Spin}$ structure is something rather rigid, and consider the fact we had uncountably many smooth structure in dimension 4, it seems quite unlikely that a $\textit{spin}^{c}$ structure would exist for all smooth four manifolds.
Not long afterwards a colleague told me this is settled; a source claimed that $\textit{spin}^{c}$ structure exists for all smooth manifolds of dimension less than or equal to 4. I was a bit surprised and decided to double check. It was not clear to me why it would work for all four dimensional manifolds, and if it works - why stop at 4? (this statement is obviously wrong for trivial reasons, as $\textit{spin}^{c}$ structure only exists on oriented manifolds).
I think I now want to ask about this question seriously. I read the wikipedia article carefully, but after tracking back the source, I do not see the book (Kirby calculus and 4 manifolds) proved any statement like that. On other hand, I think I saw on the Seiberg-Witten invariant page that $\textit{spin}^{c}$ structure exists for all smooth, compact oriented four manifolds. This is a reasonable statement to believe, but how to prove it? The wikipedia article on $\textit{spin}^{c}$ structure claimed that:
"A $\textit{spin}^{c}$ structure exists if the manifold is orientable and ....in other words, the third integral Stiefel-Whitney class vanishes)"
The proof is in the section "details". Now, I have trouble believing the all smooth, compact oriented four manifolds have $w_{3}(M)=0$. Since an axiomatic point of view is obviously not helpful, I tried to review the corresponding chapter in Hatcher, which says(bottom of page 75):
"..For each cell the obstruction to extending lies in $\pi_{2}(SO(n))$. This group happens to be $0$ for all $n$, so the section automatically extends over $B^{3}$. "
According to this, for all reasonable manifolds(not just dimension 4); $w_{3}(M)=0$. If I am not mistaken this fact $(\pi_{2})(G)=0$ if $G$ is a semisimple Lie group) is proved by Bott. However, this definition (Whitney's original definition) seems to be subtlely different from the modern definition(see the next page in Hatcher). So I want to ask if my reasoning process works through. I thought I do "know" characteristic classes, but obviously I only knew them at a superficial level that I cannot prove this fact myself without referring anything.
After consulting the usual reference (John Morgan's book), it seems this is proved in the Chapter introducing the $\textit{Spin}^{c}$ structure. However his proof involves $\mathbb{Z}/2^{k}\mathbb{Z}$ homology classes and is not very readable from my point of view. I suspect an independent proof by myself needs to be constructed. This seems standard enough that probably too low for mathoverflow.