TITLE: Using Stiefel-Whitney class to build new principal bundles
QUESTION [5 upvotes]: I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me.
Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ is a Riemann surface.

If $\mathcal{w}_2(P)=0$, then $P$ is covered by a principal $SU(2)$-bundle to which we may associate a rank-2 vector bundle $V$, with $\mathcal{c}_1(V)=0$.

My questions:
1.1) What does he mean by saying "$P$ is covered by another bundle"?
1.2) Why $\mathcal{w}_2(P)=0$ implies the existence of such a cover?

If $\mathcal{w}_2(P)\neq 0$, then there is a principal $U(2)$ bundle $\hat P$ to which $P$ is associeted via homomorphism $U(2)/Z(U(2))\simeq SO(3)$. Associeted to $\hat P$ is a rank-2 vector bundle $V$ with $\mathcal{c}_1(V)$ is odd. Fixing a connection $A_0$ on $\wedge^2V$, we find that a connection $A$ on $\hat P$ lifts to one on $P$, whose curvature is $F(A)+\frac{1}{2}F(A_0)1$.

More questions:
2.1) Again, why $\mathcal{w}_2(P)=0$ implies the existence of such a cover?
2.2) How can I find this connection $A$?
Sorry for all these questions, but I'm pretty new in this subject and I have no idea of where can I find these results. So, any explanation and reference are very welcome.

REPLY [4 votes]: This is an answer only to the question 1.2 (1.1 is already answered in a comment by Ulrich Pennig).
Consider the adjoint representation of $SU(2)$ on its Lie algebra.  Noting that the real
dimension of $SU(2)$ is 3, this gives a homomorphism $SU(2)\to SO(3)$.  One can prove 
that it is surjective with the kernel isomorphic to $Z/2$.  Thus we get a fibration sequence $$Z/2 \to SU(2) \to SO(3) \to BZ/2\to BSU(2)\to BSO(3) \to K(Z/2,2).$$
Suppose now $P$ is a principal $SO(3)$ bundle over $X$, classified by a map $f:X\to BSO(3)$.  Then $w_2(P)$ is nothing but the composition $X\stackrel{f}{\to}BSO(3)\to K(Z/2,2)$.  Thus its vanishing is equivalent to existence of $f$ to $BSO(3)$.  Take the corresponding $SO(3)$-principal bundle, and you get the desired cover.