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 ...

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 ...

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, ...

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 ...

Community wiki

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 $...

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 ...

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)...

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 ...

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 ...

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 ...

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 $f$ ...

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 "...

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 ...

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".
...

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 ...

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, ...

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 ...

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 ...

Community wiki

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 ...

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
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 ...

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 ...

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 > ...

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 ...

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 ...

Community wiki

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
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 ...

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 ...

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