I would like to know what has been done in terms of specific calculations for the cohomology of principal bundle. For instance, it is known (see e.g. Greub, Halperin and Vanstone's …

I am considering the following situation. Let $M_5$ be a 5-dimensional manifold which is an $S^1$ principal bundle over 4-manifold $M_4$. For instance, $M_5 = S^5$ and $M_4 = \ma …

What exactly do mathematicians mean when they refer to "the data" involved in a construction? I've encountered this many times and I can usually figure out what's going on, but I …

This is related to my question Adelic description of moduli of $G$-bundles on a curve. Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mat …

In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition). Does it make sense to spea …

Let $P \xrightarrow{p} B$ be a 2-$\mathcal{G}$-bundle where $P$ and $B$ are two 2-spaces (categories internalized in $\mathrm{Diff_{\infty}}$ thus forming a 2-category $\mathrm{2Ca …

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where: $G$ is a …

I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization. The Statement Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a …

While reading through Brylinski, as in all of my posts, I am trying to understand the following equation: $ g_* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$ Setting I have a …

Let $X$ be a smooth, projective, geometrically connected curve over a field $k$ and $G$ an an affine algebraic group group over $k$ (we can put more hypotheses on $G$ if necessary) …

Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X. I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus. In parti …

Let $G$ be a topological group. Recall that its classifying space $BG$ is a CW-complex which is the base of a locally trivial principal bundle of group $G$, with contractible tota …

It is a well known fact that (isomorphism classes of) principal $S^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z})$. There …

I am giving a presentation soon on the Classification of Complex Line Bundles and I would like to have some very "basic" visualizations to use as examples. Background and Context …

A principal $G$-bundle has a cross section iff it is trivial (e.g. Husemoller's Fibre Bundles, 3rd ed., 8.3 in chapter 4). A principal $G$-bundle is in particular a fiber bundle w …