Universal vector bundle on the Grassmannian

Example 10.1.9. Let us consider the Grassmannian \(\Gr _n(\R ^\infty )\) of \(n\)-dimensional subspaces of \(\R ^\infty \). By the real analogue of the Grassmannian classification theorem recorded in Remark 9.2.22, this space is a model for the classifying space \(BO(n)\) and classifies rank \(n\) real vector bundles over paracompact Hausdorff spaces. It is a cell complex by its Schubert cell decomposition. The universal bundle \(\gamma _n\) over \(\Gr _n(\R ^\infty )\) has as total space \[E_n = \{(V,v) \in \Gr _n(\R ^\infty ) \times \R ^\infty \mid v \in V\}\] with the projection map sending \((V,v)\) to \(V\). The fiber over a point \(V \in \Gr _n(\R ^\infty )\) is precisely the \(n\)-dimensional vector space \(V\) itself. The Thom space \(\Th (\gamma _n)\) is the space of pairs \((V,v)\) where \(v \in V\) has length \(\leq 1\), modulo those pairs where \(v\) has length exactly \(1\).