# Tag Info

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### 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
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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 ...
• 25.9k

### 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

### 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 ...
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• 40.4k

### 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

### 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
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### 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
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### 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
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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 "...
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### 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

### 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
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### 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
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### 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, ...
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### 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 ...
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### 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
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### 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 ...

### 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 ...

### 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 ...
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### 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 $F$. ...
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### 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