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 ...
- 68.1k
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 ...
- 27.4k
17
votes
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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 "...
- 399
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 ...
- 7,627
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 ...
- 27.3k
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 $...
- 55.4k
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 ...
- 34.4k
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 ...
- 6,809
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 ...
- 16.4k
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".
...
- 26.9k
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 ...
- 2,197
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 ...
- 151
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 ...
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 ...
- 27.3k
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 > ...
- 6,526
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 ...
- 10.4k
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, ...
- 27.3k
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 ...
- 55.4k
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 ...
- 26.1k
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. ...
- 118k
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 ...
- 2,820
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 ...
- 178
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 ...
- 2,022
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 $E \to E \times_X E$.
- 38.5k
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)...
- 108k
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 ...
- 19.3k
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 ...
- 6,831
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 ...
- 26.1k
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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