---- > [!definition] Definition. ([[sphere bundle]]) > Let $E \xrightarrow{\pi}X$ be a [[vector bundle]] of dimension $d$ with typical fiber $\mathbb{R}^{d}$. Put $E^{\sharp}:=E-s_{0}(X)$. > Suppose $E$ is endowed [[Whitney sum of vector bundles|with]] [[inner product on a vector bundle|inner product]] $\langle -,- \rangle: E \oplus E \to \mathbb{R}$. Define the **sphere bundle $\mathbb{S}(E)$ of $E$** as $\mathbb{S}(E):= \{ v \in E: \langle v,v \rangle=1 \}.$ The [[inclusion map|inclusion]] $\mathbb{S}(E) \xhookrightarrow{j} E^{\sharp}$ is a [[homotopy equivalent|homotopy equivalence]] with [[homotopy]] inverse given by normalization.[^1] ^definition ---- #### [^1]: [[punctured Euclidean space and the unit sphere have isomorphic fundamental groups|Just like]] $\mathbb{S}^{d-1} \hookrightarrow \mathbb{R}^{d}-\{ 0 \}$. ---- #### 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 > ```