On the base manifold, the complex projective space ${\mathbb{P}}_2({\mathbb{C}})$ (or ${\mathbf{CP}}_2$), I am interested in knowing how to classify and construct explicit distinct classes of

  1. tautological line bundles $L_{\mathbb{C}}$ over ${\mathbb{P}}_2({\mathbb{C}})$

  2. principle SO(3) bundle over ${\mathbb{P}}_2({\mathbb{C}})$

and how the above two bundles (tautological line bundle and principle SO(3) bundle) have any relations/constraints to

  1. tangent bundle over ${\mathbb{P}}_2({\mathbb{C}})$,

say, through the computations of characteristic classes.

For example, I think we have (the partial info I can obtain, please double check):

$\bullet$ $c_1 (\mathbb{CP}^2) =3 \in H^2(\mathbb{CP}^2, \mathbb{Z})$.

$\bullet$ $w_2(\mathbb{CP}^2)=c_1 (\mathbb{CP}^2) \mod 2=1 \mod 2 =1 \in H^2(\mathbb{CP}^2, \mathbb{Z}_2)$.

$\bullet$ $c_2 (\mathbb{CP}^2)=3 \in H^4(\mathbb{CP}^2, \mathbb{Z})$.

$\bullet$ $p_1 (\mathbb{CP}^2)=3 \in H^4(\mathbb{CP}^2, \mathbb{Z})$.

$\bullet$ $w_2(L_{\mathbb C})=c_1(L_{\mathbb C}) \mod 2=-1 \mod 2=1 \in H^2(?, \mathbb{Z}_2)$.

Here $c_j$ stands for Chern class, $w_j$ stands for Stieffel Whitney class.

(q1) What are the questions marks above in $H^2(?, \mathbb{Z})$ and $H^2(?, \mathbb{Z}_2)$?

(q2) How do we classify 1. tautological line bundles, 2. principle SO(3) bundle over ${\mathbb{P}}_2({\mathbb{C}})$? How many isomorphism classes there are? How do we construct them? (e.g. relating the associated vector bundles to the principle SO(3) bundle)

(q3) How do we relate the principle SO(3) bundle to the tautological line bundles? (e.g. relating the tautological line bundles to the associated vector bundles, then to the principle SO(3) bundle?)

Here we may write the associated vector bundle $E = 1 \oplus L_{\mathbb{R}} \oplus \dots$ or $E = 1 \oplus L_{\mathbb{C}} \oplus \dots$ in terms of a decomposition into a set of the trivial real vector bundle 1, and the tautological real /or complex line bundle $L_{\mathbb{R}}$ /or $L_{\mathbb{C}}$ (yes ?).


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