(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
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 May 30, 2017 at 15:10:36. See the history of this page for a list of all contributions to it.