On the base manifold, the quaternion projective space ${\mathbb{P}}_1({\mathbb{H}})$ (or ${\mathbf{HP}}_1 \simeq S^4$) (homeomorphic and diffeomorphic to $S^4$), I am interested in knowing how to construct

  1. line bundles over ${\mathbb{P}}_4({\mathbb{R}})$

classify and construct explicit distinct classes of

  1. principle SO(3) bundle over ${\mathbb{P}}_4({\mathbb{R}})$

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

  1. tangent bundle over ${\mathbb{P}}_4({\mathbb{R}})$,

say, through the computations of characteristic classes.

For example, $\mathbb{HP}^1 \simeq S^4$, I think we have (the partial info I can obtain, please double check):

$\bullet$ $w_1 ({S}^4) =0$.

$\bullet$ $w_2 ({S}^4) =0$.

$\bullet$ $c_1 ({S}^4) =?$.

$\bullet$ $c_2 (S^4)=?$.

$\bullet$ $p_1 (S^4)=$$0$.

$\bullet$ $w_1 (E) =?$.

$\bullet$ $w_2 (E) =?$.

$\bullet$ $c_1 (E) =?$.

$\bullet$ $c_2 (E)=?$.

$\bullet$ $p_1 (E)=?$

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

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

(q1) What are the questions marks above?

(If we compute the characteristic classes of the tangent bundle of $S^4$, or the principle $SO(3)$ bundles $E$?)

(q2) How do we classify 1. tautological line bundles, 2. principle SO(3) bundle over ${\mathbb{P}}_1({\mathbb{H}})$? 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, say $E=2 L_{\mathbb R}+1$ as a nontrivial bundle, then to the principle SO(3) bundle?)

  • $\begingroup$ Vector bundles over $S^4$ are classified by homotopy classes of maps $S^3\rightarrow G$, where $G$ is the structure group. This is the clutching construction. This corresponds to $\pi_3(G)$. These groups are known (see for example wikipedia). You can find more information in Hatcher's lecture notes on vector bundles. Some of the characteristic classes are obviously zero since they live in zero cohomology groups. BTW: Not all vector bundles split as sums of line bundles (otherwise all characteristic classes over higher dimensional spheres would vanish!) $\endgroup$ – Thomas Rot Jul 27 '18 at 14:27
  • $\begingroup$ thanks for the great generous comments! $\endgroup$ – wonderich Jul 27 '18 at 14:44

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