----- > [!proposition] Proposition. ([[pullback of a principal bundle is trivial]]) > Let $\pi:P \to B$ be a [[principal bundle over a smooth manifold|principal]] $G$-bundle. Show that $\pi^{*}P \to P$ is a trivial principal $G$-bundle. ^proposition - [ ] d > [!proof]- Proof. ([[pullback of a principal bundle is trivial]]) > ~ > > > We will instantiate the principal $G$-bundle definition in terms of the universal property by taking, for general bundle $P \xrightarrow{\pi}B$ and smooth map $B' \xrightarrow{f}B$, the pullback to be $f^{*}P=\{ (b', p) \in B' \times E: f(b')=\pi(e) \} \subset B' \times E$ > where $G$ is understood to act on $f^{*}P$ via $(b', p)h:=(b', ph)$ for all $h \in G$. In this case the map $F:f^{*}P \to B$ from the pullback definition is the (restriction of) the canonical projection $B' \times E \to E$. > > With that in mind, define $\pi^{*}P := \{ (p_{1},p_{2}) \in P \times P : \pi(p_{1})=\pi(p_{2})\}\subset P \times P,$ > and consider the commutative diagram > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB120sA9AKgAKIAL6l0mXPkIoAjOSq1GLNkNHjseAkTIyF9Zq0QhVYkBg1Sic3dX3KjAIREKYUAObwioAGYAnCAC2SGQgOBBIAEy2SoYc7HgMsMCc3MIAvJw4MAAeOMBovsIA+hEg1Ax0AEYwDAISmtIgvlhuABY4IqZ+gcHUYUhyigZsAGIZ8Tl5BcUynT7+QYiD-YgAzNHDRilYcyDdi1Gh4Wsb9nvj3GUgFdW19ZZGzW0dwhTCQA > \begin{tikzcd} > \pi^*P \arrow[d, "\tilde{\pi}=\text{pr}_1"'] \arrow[r, "F=\text{pr}_2"] & P \arrow[d, "\pi"] \\ > P \arrow[r, "f=\pi"'] & B > \end{tikzcd} > \end{document} > ``` > note that for all $h \in G$ one has $F\big( (p_{1},p_{2})h \big)=F(p_{1},p_{2}h)=p_{2}h=F(p_{2})h$ and so this is indeed a pullback. That $\pi^{*}P$ is trivial is witnessed by the existence of a smooth global section $\begin{align} > s:P &\to \pi^{*}P \\ > p & \mapsto (p, p) > \end{align}$ > and appealing to question 8. ^0f90f0 ----- #### ---- #### 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 > ```