0
votes
0answers
88 views
State of the game : cohomology of principal bundles
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 …
0
votes
1answer
142 views
How to characterize this particular kind of bundle?
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 …
4
votes
5answers
345 views
What is “Data” involved in a mathematical construction?
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 …
3
votes
1answer
272 views
Trivializing principal bundles on a curve over a finite field
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 …
2
votes
1answer
161 views
Confusions over the definitions of universal bundle and characteristic class
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 …
0
votes
0answers
60 views
Deriving local data for 2-transition function of a 2-bundle (getting stuck)
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 …
3
votes
3answers
269 views
What are the symmetries of a principal homogeneous bundle?
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 …
2
votes
1answer
128 views
Twisting an object P by an H-Torsor I
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 …
4
votes
3answers
311 views
Connection Transformation Formula; Degree 3 Cech Cohomology
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 …
5
votes
3answers
426 views
Adelic description of moduli of $G$-bundles on a curve
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) …
3
votes
1answer
229 views
Higgs bundle and stable bundle
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 …
2
votes
1answer
181 views
Numerable covers from the point of view of Grothendieck topologies
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 …
3
votes
4answers
613 views
Nontrivial examples of non-trivial principal circle bundles
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 …
16
votes
2answers
703 views
How to Draw Complex Line Bundles
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
…
2
votes
1answer
363 views
Cross sections in bundles and principal G-bundles
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 …

