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You are correct that locally, there is no difference between SU(2) and SO(3) bundles, but there are important global differences. In particular, SO(3) bundles may have non-trivial $w_2$, which obstructs their lifting to an SU(2) bundle. The Dold-Whitney theorem (Classification of oriented sphere bundles over a 4-complex. Ann. of Math. (2) 69 1959 667–677) says that the topological classification of SO(3) bundles is given exactly by $w_2$ and the first Pontrjagin class $p_1$. The only constraint on the two classes is that $p_1$ is the (mod 4 valued) Pontrjagin square of $w_2$; compare The Stiefel-Whitney and Pontryagin classes of SO(3)-bundlesThe Stiefel-Whitney and Pontryagin classes of SO(3)-bundles. There are also some subtle differences in the topology of the gauge group.

In mathematical gauge theory, as applied to 4-manifold topology, this is an important distinction. In many circumstances, one can prove that the moduli space of anti-self-dual SO(3) connections is compact, leading to somewhat simpler proofs of some of Donaldson's fundamental theorems. This was observed by Fintushel and Stern (SO(3)-connections and the topology of 4-manifolds, J. Differential Geom. 20 (1984), 523-539).

You are correct that locally, there is no difference between SU(2) and SO(3) bundles, but there are important global differences. In particular, SO(3) bundles may have non-trivial $w_2$, which obstructs their lifting to an SU(2) bundle. The Dold-Whitney theorem (Classification of oriented sphere bundles over a 4-complex. Ann. of Math. (2) 69 1959 667–677) says that the topological classification of SO(3) bundles is given exactly by $w_2$ and the first Pontrjagin class $p_1$. The only constraint on the two classes is that $p_1$ is the (mod 4 valued) Pontrjagin square of $w_2$; compare The Stiefel-Whitney and Pontryagin classes of SO(3)-bundles. There are also some subtle differences in the topology of the gauge group.

In mathematical gauge theory, as applied to 4-manifold topology, this is an important distinction. In many circumstances, one can prove that the moduli space of anti-self-dual SO(3) connections is compact, leading to somewhat simpler proofs of some of Donaldson's fundamental theorems. This was observed by Fintushel and Stern (SO(3)-connections and the topology of 4-manifolds, J. Differential Geom. 20 (1984), 523-539).

You are correct that locally, there is no difference between SU(2) and SO(3) bundles, but there are important global differences. In particular, SO(3) bundles may have non-trivial $w_2$, which obstructs their lifting to an SU(2) bundle. The Dold-Whitney theorem (Classification of oriented sphere bundles over a 4-complex. Ann. of Math. (2) 69 1959 667–677) says that the topological classification of SO(3) bundles is given exactly by $w_2$ and the first Pontrjagin class $p_1$. The only constraint on the two classes is that $p_1$ is the (mod 4 valued) Pontrjagin square of $w_2$; compare The Stiefel-Whitney and Pontryagin classes of SO(3)-bundles. There are also some subtle differences in the topology of the gauge group.

In mathematical gauge theory, as applied to 4-manifold topology, this is an important distinction. In many circumstances, one can prove that the moduli space of anti-self-dual SO(3) connections is compact, leading to somewhat simpler proofs of some of Donaldson's fundamental theorems. This was observed by Fintushel and Stern (SO(3)-connections and the topology of 4-manifolds, J. Differential Geom. 20 (1984), 523-539).

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Danny Ruberman
Danny Ruberman
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You are correct that locally, there is no difference between SU(2) and SO(3) bundles, but there are important global differences. In particular, SO(3) bundles may have non-trivial $w_2$, which obstructs their lifting to an SU(2) bundle. The Dold-Whitney theorem (Classification of oriented sphere bundles over a 4-complex. Ann. of Math. (2) 69 1959 667–677) says that the topological classification of SO(3) bundles is given exactly by $w_2$ and the first Pontrjagin class $p_1$. The only constraint on the two classes is that $p_1$ is the (mod 4 valued) Pontrjagin square of $w_2$; compare The Stiefel-Whitney and Pontryagin classes of SO(3)-bundles. There are also some subtle differences in the topology of the gauge group.

In mathematical gauge theory, as applied to 4-manifold topology, this is an important distinction. In many circumstances, one can prove that the moduli space of anti-self-dual SO(3) connections is compact, leading to somewhat simpler proofs of some of Donaldson's fundamental theorems. This was observed by Fintushel and Stern (SO(3)-connections and the topology of 4-manifolds, J. Differential Geom. 20 (1984), 523-539).