48
votes
Accepted
Are there "principal" bundles $S^1 \to S^3 \to S^2$ other then Hopf's? (They would be necessarily not locally trivial)
The 3-sphere has infinitely many Seifert fibrations with generic
fiber a torus knot (including the unknot).
For a $(p,q)$ torus knot, the Hopf invariant will be $pq$ (up to sign).
To see this, note ...
- 61.9k
20
votes
Accepted
For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
I like to think of $EG$ and $BG$ in terms of configuration spaces.
The space $BG$ can be identified with the following configuration space. It consists of configurations of finitely many points in the ...
- 25.9k
19
votes
Generalising the Penrose Twistor Fibration
Yes, there is such a twistor fibration over each $S^{2n}$, and the resulting manifold is a complex manifold endowed with a holomorphic $n$-plane field transverse to the fibers of the mapping. Namely, ...
- 98.4k
15
votes
Vector bundles vs principal $G$-bundles
I believe you have answered this yourself. Principal bundles are trivial iff they admit a global smooth section. Vector bundles always admit a global smooth section (zero section).
Therefore vector ...
15
votes
Accepted
Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?
No. (The main idea here is present in Dylan Wilson's comment.)
Every principal $SU(2)$-bundle over $S^2$ is trivial, because $\pi_1 SU(2)$ is trivial. But there is a nontrivial oriented bundle over $...
- 48.7k
15
votes
Is there a book on differential geometry that doesn't mention the notion of charts?
For Riemannian geometry (and therefore no gauge theory or Hamiltonian mechanics), I recall two beautiful coordinate-free expositions:
1) Milnor's monograph "Morse Theory" has all of the essentials ...
- 24.9k
14
votes
Principal bundles that can't be detected by spheres
The simplest example of a principal bundle that can't be detected by spheres:
The $S^1$-bundle over the 2-dimensional torus $T^2$ whose total space is the quotient $Heis_3(\mathbb R)/Heis_3(\mathbb Z)...
- 40.4k
14
votes
What are the applications of the Atiyah-Bott Yang Mills paper?
They observed that the algebraic concept of stability is equivalent with the analytic concept of Yang-Mills connection. This has a variational characterization opening the door for the usage ...
- 32.3k
13
votes
Topological obstruction for the existence of spin$^c$ structure
The obstruction to spin$^c$ is often denoted by $W_3(M) \in H^3(M,\mathbb{Z})$. It is obtained as follows: Let
$$
\beta \colon H^2(M,\mathbb{Z}/2\mathbb{Z}) \to H^3(M,\mathbb{Z})
$$
be the Bockstein ...
- 7,277
13
votes
Accepted
Mednykh's formula for 2d manifold with boundaries? How to count principal G-bundles with prescribed holonomies?
I get a slightly different formula:
$$\sum_{V} \frac{d(V)^{2g-2-n}}{|G|^{2g-2}} \prod_{i=1}^n |k_i|\chi_V(k_i)$$
Here $|k_i|$ denotes the size of the conjugacy class.
I prefer to express this in a ...
- 6,394
12
votes
Accepted
Principal bundles that can't be detected by spheres
If all maps are involved are pointed, your question is equivalent to the following: if $f : X \to BG$ is a map (where $X$ is connected) such that the induced map on $\pi_{\bullet}$ is zero, is $f$ ...
- 108k
11
votes
Accepted
An intuitive explanation for group cohomology via cochains?
What I'm going to say is pretty much the same that JK34 has written in their answer, but in a more elementary approach that is hopefully adding some insight.
Suppose that you want to look at the "...
- 339
10
votes
Accepted
Classifying spaces and Brown's representability theorem
No, homotopy classes of maps of unpointed spaces are not equivalent to homotopy classes of maps of pointed spaces. You need to be a little bit more clever. What follows is a spelling out of Brown's ...
- 15.7k
10
votes
Classification of bundles, Postnikov towers, obstruction theory, local coefficients
I'll try to answer question 1. Unfortunately, I know of no especially convenient reference for the case of understanding general Postnikov towers; however, everything I say below is "well known".
...
- 25.8k
10
votes
Accepted
A nontrivial principal bundle which satisfies Leray-Hirsch theorem
Let $P$ be any $SU(2)$-bundle on $X$ with vanishing second Chern class $c_2(P)$. The hypotheses of the Leray-Hirsch theorem are satisfied if there is a class in $H^3(P)$ which restricts to the ...
- 1,238
9
votes
Accepted
Lindel's theorem for semisimple simply connected G
In full generality, the answer is no. There are examples of Parimala of non-extended torsors for special orthogonal groups over $\mathbb{A}^2_{\mathbb{R}}$, see e.g. Amer J. Math. 100 (1978), 913-924, ...
- 16.6k
9
votes
What are global sections of the determinant bundle on the Beilinson-Drinfeld Grassmannian?
In a recent paper https://arxiv.org/abs/2003.12930 the space of global sections on Schubert subvarietis of Beilinson-Drinfeld Grassmanian was computed. It turns out to be global Demazure module over ...
9
votes
Accepted
Circle Action on Quaternionic Projective Space
$\mathbb{HP}^n\cong \mathrm{Sp}(n+1)/(\mathrm{Sp}(n)\times \mathrm{Sp}(1))$ is a symmetric space, so every one-parameter subgroup of $\mathrm{Sp}(n+1)$ acts on it. As was noted in the comments, such ...
- 4,434
9
votes
Accepted
Is there a group scheme G such that there is a proper class of non-isomorphic fpqc G-bundles over some fixed base?
In more down-to-earth terms, you are asking whether there is always a set of $G$-torsors for the fpqc topology over a scheme $S$ such any every $G$-torsor for the fpqc topology over $S$ is isomorphic ...
9
votes
How does one introduce characteristic classes
I would follow Chern in his short paper Vector bundles with a connection, where he introduces characteristic classes as polynomials in the curvature of a connection, and then shows (easily) their ...
8
votes
Generalising the Penrose Twistor Fibration
One possible generalization of Penrose twistor fibration is the fibration $\pi:\mathcal{J}_M\to M$ over any oriented $2n$-dimensional conformal manifold $(M,[g])$, whose fiber $\pi^{-1}(x)$ is the ...
- 9,069
8
votes
Automorphism group of a fiber bundle surjects onto diffeomorphism group?
Any fiber bundle $\pi:P \longrightarrow M$ with fiber $F$ is classified by a map $f_{\pi}: M \longrightarrow B Diff(F)$ where $B Diff(F)$ is the classifying space of the diffeomorphism group of $F$. ...
- 8,856
8
votes
What are the applications of the Atiyah-Bott Yang Mills paper?
Hitchin 1987 extended the work of Atiyah-Bott to study the topology of the moduli space of Higgs bundles on $\Sigma$ via the Yang-Mills functional. Simpson 1988 proved that this moduli space agrees ...
- 9,916
8
votes
Alternative (easier) Proof of Ambrose Singer Holonomy theorem
I always find it easier to work with the vector bundle induced by a linear representation of the structure group. I believe this theorem is a consequence of the following loop formula (a terse proof ...
- 24.9k
8
votes
A nontrivial principal bundle which satisfies Leray-Hirsch theorem
Suppose $M = S^n$ is a sphere with $n$ odd and at least $5$. Pick your favorite Lie group $G$ for which $\pi_{n-1}(G)$ is non-trivial. (Many examples may be found here. For example, for any $n > ...
- 5,756
8
votes
Accepted
Classifying space of bundles over bundles
If I understand the question right, I think the classifying space can be described like this. Let $Map(G,BH)$ be the space of all continuous maps from $G$ to $BH$. Make the group $G$ act continuously ...
- 48.7k
7
votes
Vector bundles vs principal $G$-bundles
This is an old question, but I was just thinking about this question myself and I came to this realization:
For a vector bundle over a field $K$ of rank $n$, if it were a principal bundle, it would ...
7
votes
Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?
I'll try to say as much as I can about each of the four questions above.
If $X$ is a space with an action of a topological group $G$, then the quotient map $X\to X/G$ is a principal bundle if and ...
- 7,876
7
votes
Accepted
What does "higher monodromy" tell us about a principal bundle
There is a monodromy of sorts for any topological bundle (or even fibration) $\pi \colon E \to X$ with fiber $F$. Let $\gamma \colon [0,1] \to X$ be a path and set
$F = \pi^{-1}(\gamma(0))$. The ...
- 794
7
votes
Motivation for construction of associated fiber bundle from a principal bundle
To answer your specific question, you get a principal $H$-bundle, often denoted $P \times^G H$. The transition maps are clearly just given by applying $\phi$ to those of $P$.
I think you want a ...
- 23.4k
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