Principal bundle over S1 from a clutching function

This is the $1$ -dimensional case of the general clutching construction for bundles.

Let $G$ be a Lie group, and consider the circle as a quotient $S^1 \cong [0,1]/(0\sim 1)$ .

Construction (gluing by an element of G)

Fix an element $g\in G$ . Define

P_g := \big([0,1]\times G\big)\big/\sim,

where the equivalence relation identifies the ends by

(0,h)\sim (1,gh)\qquad\text{for all }h\in G.

Let $\pi:P_g\to S^1$ be induced by the projection $[0,1]\times G\to [0,1]\to S^1$ , and let $G$ act on $P_g$ by the right action

[(t,h)]\cdot k := [(t,hk)].

Then $\pi:P_g\to S^1$ is a principal G-bundle.

One can describe this equivalently in transition-function language: choose an open cover of $S^1$ by two arcs $U_0,U_1$ so that $U_0\cap U_1$ has two connected components, and take transition functions that are constant on each component, with one of them equal to $g$ . This viewpoint ties the construction to principal bundle transition functions and the cocycle condition.

Isomorphism remarks

If $G$ is connected, then every principal $G$ -bundle over $S^1$ is trivial (since $[S^1,BG]\cong \pi_1(BG)\cong \pi_0(G)$ ). In that case, each $P_g$ is isomorphic to the trivial principal bundle even when $g\neq e$ .
If $G$ is discrete (or more generally not connected), different clutching data can produce non-isomorphic bundles. Over $S^1$ , the classification reduces to homomorphisms $\pi_1(S^1)\cong \mathbb Z\to G$ up to conjugation, i.e. conjugacy classes of elements of $G$ , matching the intuition that $g$ records a “monodromy.”

Examples

Discrete group G = Z2: trivial vs nontrivial double cover. For $G=\mathbb Z_2=\{\pm 1\}$ , the construction gives two isomorphism classes: $P_{+1}$ (trivial) and $P_{-1}$ (nontrivial). The nontrivial bundle corresponds to the connected double cover $S^1\to S^1$ , $z\mapsto z^2$ , viewed as a principal $\mathbb Z_2$ -bundle.

O(1) bundles and the Möbius line bundle. Since $O(1)\cong \mathbb Z_2$ , the two principal $O(1)$ -bundles over $S^1$ are exactly the ones above. The nontrivial principal $O(1)$ -bundle yields, via the standard action on $\mathbb R$ , the Möbius real line bundle as an associated vector bundle (compare associated vector bundle).

Connected groups: U(1) gives only the trivial principal bundle over S1. For $G=U(1)$ (connected), any choice of $g\in U(1)$ produces a bundle $P_g$ that is isomorphic to $S^1\times U(1)$ . This is a concrete instance of the equivalence “bundle is trivial iff it admits a global section,” as in the triviality criteria for principal bundles.