49
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 ...

17
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 "...

16
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 ...

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 ...

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

### 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 ...

14
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:
$$\sum_{V} \frac{d(V)^{2-2g-n}}{|G|^{2-2g}} \prod_{i=1}^n |k_i|\chi_V(k_i)$$
Here $|k_i|$ denotes the size of the conjugacy class.
I prefer to express this in a ...

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

### 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

### 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

### 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

9
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 ...

9
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

### 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

Accepted

### Motivation for construction of associated fiber bundle from a principal bundle

This construction reverses the construction of the frame bundle from a vector bundle (e.g., the tangent bundle). The idea is that each point $f \in F_p$ in the frame bundle of a vector bundle $E$ is, ...

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

### 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 ...

7
votes

Accepted

### Characteristic classes of the bundle of trace free, skew adjoint endomorphisms

As a real vector bundle, $E^{\ast} \otimes E$ decomposes as the direct sum of two copies of the bundle $\mathfrak{su}(E)$ of trace-free skew-adjoint endomorphisms and two copies of the trivial bundle. ...

7
votes

Accepted

### Extension of Hopf fiber bundle to (an equivariant) 2 dimensional vector bundle

I think both of your questions can by answered positive. Since the Hopf fibration is an $S^1$ bundle it comes as a sphere bundle of complex line bundle over $S^2$. From the Gysin sequence the first ...

7
votes

Accepted

### When a free action gives rise to a $G$-principal bundle

For general topological spaces, I found some answers in Tammo to Dieck, "Algebraic Topology" [TD], Chapter 14.
I give here a quick summary.
Define a principal G-bundle $p$ as a continuous ...

7
votes

Accepted

### Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

I'll assume that the vector space "$V$" occuring in constructions (1) and (2) doesn't have to be the same. In that case I'll rename vector space in construction (2) to "$W$."
Then ...

6
votes

Accepted

### Global trivialization of a Principal G bundle

Yes, a principal $G$-bundle is trivial if and only if it has a section, and if you take $\tilde{X} = E$, then $f^*E = E \times_X E$ has a section given by the diagonal map $E \to E \times_X E$.

6
votes

Accepted

### The bundle of symmetric affine connections as quotient of the second-order frame bundle

There is an 'identification', i.e., a way to interpret a torsion-free affine connection on $M$ as a section of $\check J^2(M,\mathbb{R}^n_0)/\mathrm{GL}_n(\mathbb{R})$ in such a way that every (smooth)...

6
votes

### What are the applications of the Atiyah-Bott Yang Mills paper?

With 562 citations on Mathscinet, it's hard to summarize all of the applications of this influential paper! One important one was the extension of the Atiyah-Bott results to the setting of parabolic ...

6
votes

### What does reduction of structure group of principal bundle say?

Let $G$ be a topological group and $M$ be a smooth manifold. Then, a reduction of the structure group of the frame bundle from $\mathrm{GL}_n(\mathbb R)$ to $\mathrm{GL}_n(\mathbb R)\times G$ is ...

6
votes

### Alternative (easier) Proof of Ambrose Singer Holonomy theorem

You can find a complete proof, which I think is about as easy (for me) as I can manage, in the current version of my never-ending Introduction to Cartan Geometries. The proof still involves the ...

6
votes

Accepted

### Connecting torsors by a rational curve

Welcome new contributor. That is true, even for $G$ a geometrically reductive group scheme, not merely for finite groups. The proof is elementary, but it is a bit long. It is related to the ...

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