---- > [!definition] Definition. ([[transition functions for a principal bundle over a smooth manifold]]) > Let $B$ be a [[smooth manifold]] and $G$ a [[Lie group]] [[group action|acting]] [[smooth group action|smoothly]] and [[free group action|freely]] on the right of $B$. > Let $P \xrightarrow{\pi} B$ be a $G$-[[principal bundle over a smooth manifold|principal bundle]]. Consider two local trivializations $(U_{\alpha}, \Phi_{\alpha})$, $(U_{\beta}, \Phi_{\beta})$ with nontrivial intersection $U_{\alpha} \cap U_{\beta}$. The map $\Phi_{\beta}\circ \Phi_{\alpha}^{-1}$ on $(U_{\alpha} \cap U_{\beta}) \times G$ behaves as $\Phi_{\beta} \circ \Phi_{\alpha}^{-1}(b, g)=(b , \psi_{\beta \alpha} (b,g)) \ (*)$ for some map $\psi_{\beta \alpha}$. Considering the map $\psi_{\beta \alpha}(b, \cdot):G \to G$, one has from **(2)** in the [[principal bundle over a smooth manifold|principal bundle]] definition that $\psi_{\beta \alpha}(b, gh)=\psi_{\beta \alpha}(b,g)h$ [^1] for all $g,h \in G$. > Therefore, setting as notation $\psi_{\beta \alpha}(b):= \psi (b, 1_{G})$, we get $\psi_{\beta \alpha}(b,g)=\psi_{\beta \alpha}(b)g$ for all $b \in U_{\alpha} \cap U_{\beta}$, and may hence rewrite $(*)$ as $\Phi_{\beta} \circ \Phi_{\alpha}^{-1}(b,g)=(b, \psi_{\beta \alpha}(b)g), \ \ \psi_{\beta \alpha}:U_{\alpha} \cap U_{\beta} \to G.$ In analogy with [[transition functions for a vector bundle over a smooth manifold]], we call the $\psi_{\beta \alpha}$ **transition functions**. They satisfy the same [[transition functions for a vector bundle over a smooth manifold|cocycle conditions]]. ^definition - [ ] steenrod for constructing principal bundles [^1]: To be precise, start with the knowledge that $\Phi_{\beta} \circ \Phi_{\alpha}^{-1}(b,gh)=\Phi_{\beta} \big( \Phi_{\alpha}^{-1}(b,g)h\big)=\Phi_{\beta} \circ \Phi_{\alpha}^{-1}(b,g)h=(b, \psi_{\beta \alpha}(b,g)h)$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```