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 ...
Ian Agol's user avatar
  • 66.8k
20 votes
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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 ...
Chris Schommer-Pries's user avatar
16 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 "...
Paolo Perrone's user avatar
15 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 ...
Ulrich Pennig's user avatar
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 $...
Tom Goodwillie's user avatar
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 ...
Deane Yang's user avatar
  • 26.9k
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 ...
Liviu Nicolaescu's user avatar
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 ...
Sam Gunningham's user avatar
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 ...
Denis Nardin's user avatar
  • 16.2k
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". ...
Charles Rezk's user avatar
  • 26.6k
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 ...
Joshua Mundinger's user avatar
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
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 ...
Bertram Arnold's user avatar
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 ...
Ilya Dumanski's user avatar
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 ...
Deane Yang's user avatar
  • 26.9k
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 ...
ThiKu's user avatar
  • 10.3k
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, ...
Deane Yang's user avatar
  • 26.9k
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 > ...
Jason DeVito - on hiatus's user avatar
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 ...
Tom Goodwillie's user avatar
7 votes

Counting isomorphism classes in open subsets of Bun_G

I believe that the claim is true, and that this is related to the so-called 'very good' property of the stack $\mathrm{Bun}_G$. Following [Beilinson, Drinfeld, Quantization of Hitchin's integrable ...
t3suji's user avatar
  • 4,450
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 ...
Ben McKay's user avatar
  • 25.4k
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. ...
Qiaochu Yuan's user avatar
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 ...
Panagiotis Konstantis's user avatar
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 ...
kiran's user avatar
  • 1,982
6 votes
Accepted

group actions on fibre bundles

No. let $F$ have a $G$-action, take $B=EG$ and $E=EG \times F$ with the diagonal action. Then $\eta$ is the Borel construction $EG \times_G F \to BG$ so need not be trivial.
Oscar Randal-Williams's user avatar
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)...
Robert Bryant's user avatar
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 $E \to E \times_X E$.
Sasha's user avatar
  • 37k
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 ...
Danny Ruberman's user avatar
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 ...
Arun Debray's user avatar
  • 6,766

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