(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 morphisms (continuous function) $f \colon X \to B GL(n)$ which is unique up to homotopy:
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 $E O(n) \to B O(n)$:
Similarly for complex vector bundles
etc.
For $n, k \in \mathbb{N}$, and $n \leq k$, there is the Grassmannian manifold given as the coset topological space
Similarly, the Stiefel manifold is the coset
The quotient projection
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.
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
Last revised on January 24, 2021 at 10:20:55. See the history of this page for a list of all contributions to it.