The hopf bundle over $S^1$ is the bundle obtained after two twists. I was wondering if this bundle is the trivial bundle. Intuitively it seems like it should not be trivial since there are two twists but it seems like there exists a non zero section. So I do not know what is going wrong. If anyone can help it would be great. Thanks.
2026-04-03 01:28:12.1775179692
The hopf bundle over $S^1$ is a trivial bundle.
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The hopf bundle is a one dimensional vector bundle and so since has a section, as you noted, it is automatically trivial. What is perhaps confusing is that the picture we normally think of is its embedding into $\mathbb{R}^3$. This embedding is not isotopic to the normal embedding of $S^1 \times [0,1]$. To see this, note that the boundary of the hopf bundle is the hopf link wheras the boundary of the ordinary embedding of $S^1 \times [0,1]$ is the unlink. Nevertheless, as abstract topological spaces they are diffeomorphic.