+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Bundles +-- {: .hide} [[!include bundles - contents]] =-- #### Cohomology +--{: .hide} [[!include cohomology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition A **circle bundle** is a [[principal bundle]] for the [[circle group]] $S^1$. Equivalently this is a $U(1)$-principal bundle, for the [[unitary group]] $U(1)$. Under the canonical [[representation]] $\mathbf{B}U(1) \to Vect_{\mathbb{C}}$ the corresponding [[associated bundle]] is a complex [[line bundle]]. ## Examples * A basic example of a nontrivial [[circle group]]-[[principal bundle]] is the [[Hopf fibration]] $S^3 \to S^2$. ## Related concepts * [[circle bundle with connection]], [[regular contact manifold]] * [[prequantum circle bundle]] * [[line bundle]] * [[Seifert 3-manifold]] [[!redirects circle bundles]]