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  1. I am interested in a reference with a detailed (as simple and topological as possible) proof of the following fact:

Theorem. A principal $\mathrm{SO}(3)$-bundle on a compact oriented 4-manifold are determined by its second Stiefel-Whitney class and first Pontryagin class.

This is Theorem E.8 (Appendix) in Uhlenbeck, Freed - Instantons and 4-Manifolds (1984, 2-ed 1991), but I have difficulty following this text. For example, in the arguments preceding the theorem:

  1. Why is the universal covering of $\mathrm{SO}(3)$ called the adjoint representation? UPD: I figured out the answer to this question: the covering $\mathrm{SU}(2) \to \mathrm{SO}(3)$ is exactly the adjoint action of the unit quaternions on the pure imaginary quaternions.

  2. How are weights and characteristic classes related?

I tried to browse the reference A. Borel, F. Hirzebruch - Characteristic Classes and Homogeneous Spaces (1958) given there, but so far I have not been able to understand what it is about (another reminder to never reference without citing a specific place).

I suspect that even if I can understand the answers to these questions, I will quickly get stuck again.

  1. Any recommendations for more modern texts covering these topics?

This theorem was discussed on MO, but there were no more modern references mentioned.

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  • $\begingroup$ What do you mean, precisely, by 'modern reference'? $\endgroup$
    – mme
    Commented Dec 5 at 17:49
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    $\begingroup$ @mme I meant just expositions written later, which might be more complete/readable. I should have written the word "more" to be clearer, yes. Fixed. $\endgroup$
    – Arshak Aivazian
    Commented Dec 5 at 18:24
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    $\begingroup$ Another possible reference, but also from around the time of Borel-Hirzebruch is Dold and Whitney "Classification of oriented sphere bundles over a 4-complex", Ann. Math. 69, 1959. The proof is obstruction-theoretic (and for more general complexes, not just manifolds). Maybe a bit more focused on the concrete SO(3)-bundle classification problem than Borel-Hirzebruch. Dold and Whitney reference Pontryagin 1945 for an earlier classification result for SO(3)-bundles. $\endgroup$
    – Matthias Wendt
    Commented Dec 5 at 19:01

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