nLab universal vector bundle

Idea

In topology a universal vector bundle of some rank $n$ is a vector bundle $\zeta_n \to B GL(n)$ (over a base space to be called a classifying space) such that every other vector bundle $E \to X$ of rank $n$ over a suitably nice topological space (paracompact topological space) arises as the pullback bundle $E \simeq f^\ast \zeta_n$ of the universal bundle, along some morphism (continuous function) $f \colon X \to B GL(n)$ which is unique up to homotopy:

$$\array{ \zeta_n &\longrightarrow& E GL(n) \\ \downarrow &(pb)& \downarrow \\ X &\underset{f}{\longrightarrow}& B GL(n) } \,.$$

The universal real vector $\zeta_n$ of rank $n$ is the vector bundle which is associated to the universal principal bundle $E GL(n) \to B GL(n)$ (with structure group the general linear group) over the given classifying space, equivalently to $E O(n) \to$$B O(n)$:

$$\zeta_n \coloneqq (E O(n))\underset{O(n)}{\times} \mathbb{R}^n \,.$$

Similarly for complex vector bundles for $E U(n) \to$$B U(n)$:

$$\zeta^{\mathbb{C}}_n \coloneqq (E U(n))\underset{U(n)}{\times} \mathbb{R}^{2n} \,.$$

etc.

Constructions

Via Grassmannians and Stiefel manifolds

For $n, k \in \mathbb{N}$, and $n \leq k$, there is the Grassmannian manifold given as the coset topological space

$$Gr_n(k) \coloneqq O(k)/(O(n)\times O(k-n)) \,.$$

Similarly, the Stiefel manifold is the coset

$$V_n(k) \coloneqq O(k)/O(n) \,.$$

The quotient projection

$$V_{k-n}(k)\longrightarrow Gr_n(k)$$

is an $O(n)$-principal bundle, with associated bundle $V_n(k)\times_{O(n)} \mathbb{R}^n$ a vector bundle of rank $n$. In the limit (colimit) that $k \to \infty$ is this gives a presentation of the $O(n)$-universal principal bundle and of the universal vector bundle of rank $n$, respectively.. The base space $Gr_n(\infty)\simeq_{whe} B O(n)$ is the classifying space for $O(n)$-principal bundles and rank $n$ vector bundles.

Examples

• universal complex line bundle

• tautological line bundle

References

Textbook accounts include

• Stanley Kochmann, section 1.3 of of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Allen Hatcher, Vector bundles and K-Theory, (partly finished book) web