Universal bundles and classificant maps.

Question

We have this theorem: Let $\xi=(E,p,X)$ be a vector fiber bundle with rank($\xi)=n$. We can find a map $f: X \rightarrow Gr_n(\mathbb{C}^{\infty})$ such that $\xi=f^*(\gamma_n)$. I denote with $Gr_n(\mathbb{C}^{\infty})$ the infinite grassmannian of $n$-subspaces in $\mathbb{C}^{\infty}$ and with $f^*(\gamma_n)$ the pull-back via $f$ of the tautological bundle over $Gr_n(\mathbb{C}^{\infty})$. I have some problem to use in concrete this theorem. For example if we consider the tangent bundle over $S^2$ ($\tau_{S^2}$), how can I find $f: S^{2} \rightarrow Gr_2(\mathbb{C}^{\infty})$ such that $\tau_{S^2}=f^*(\gamma_2)$? Can you give me some other examples in order to explain a method to find this function?

Answer 1

The construction goes as follows: If $E\rightarrow B$ is a rank k vector bundle over a compact base space $B$, then, for some $n\geq k$, there is a fiberwise injective bundle homomorphism $\iota\colon E\rightarrow B\times R^n$. Having this, one can define $f\colon B\rightarrow Gr_{k}(R^n)$ by setting $f(b) := \text{image}(\iota_b: E_b\rightarrow \{b\}\times R^n)$.

In the case of $TS^n\rightarrow S^n$, we have the fiberwise injection $$ TS^n\rightarrow TS^n\oplus N(S^n\subset R^{n+1})\cong S^n\oplus R^{n+1}, $$

which gives rise to the map $S^n\rightarrow Gr_n(R^{n+1})$, $v\mapsto \{w\in R^{n+1}| v\perp w\}$.