---- > [!definition] Definition. ([[Whitney sum of vector bundles]]) > Let > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFEQBfU9TXfIRQBGUsKq1GLNgA1uvEBmx4CRAEzkJ9Zq0QcA5NwkwoAc3hFQAMwBOEALZIyIHBCSjJOtgB1vaLCDUDHQARjAMAAr8KkIgNlimABY48tZ2jogaLm6IHtrSer7+hlwUXEA > \begin{tikzcd} > E \arrow[rd, "\pi"'] & & E' \arrow[ld, "\pi'"] \\ > & X & > \end{tikzcd} > \end{document} > ``` > be [[vector bundle|vector bundles]] over a [[topological space]] $X$. Their **Whitney sum** is the vector bundle $E \oplus E'$ satisfying $(E \oplus E')_{x}=E_{x} \oplus E'_{x}$. Explicitly, $E \oplus E'$ is defined > $\begin{align} > E \oplus E' := \{ (e,e') \in E \times E' : \pi(e)=\pi'(e') \}& \to X \\ > (e,e') & \mapsto \pi(e)=\pi(e'). > \end{align}$ > Certainly $(E \oplus E')_{x}=\{ (e,e') : e \in \pi ^{-1}(x), e' \in \pi' ^{-1}(x) \}=E_{x} \oplus E'_{x}.$ ---- #### ---- #### 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 > ```