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Using Stiefel-Whitney class to build a new principal bundles

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Martin Sleziak
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Using StifelStiefel-Whitney class to build a new principal bundles

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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 curvature 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.

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 curvature 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.

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.

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user43326
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