# Questions tagged [fibre-bundles]

for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.

334
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### Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...

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### Connection of principal fiber bundles — history

I wonder who was the first to discover the notion of principal fiber bundle and its connection (gauge field in the physical language). Wikipedia cites the book by Steenrod (1951). But was he the ...

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### Non compact Seifert manifolds

A Seifert manifold $M$ is a $3$-dimensional orientable smooth manifold with an effective circle action with no fixed points.
Closed connected Seifert manifolds are classified up to an equivariant ...

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### Isomorphism between tangent bundle of $S^2$ and the kernel of a bundle homomorphism

Let $S^{4n+3} \to \mathbb{H}P^n$ be the standard projection which is a fiber bundle with fiber $S^3$. By the action of $S^1$ on $S^3$ we get a fiber bundle
$$
\mathbb{C}P^1 \xrightarrow{\iota} \mathbb{...

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### Flat Maurer-Cartan connection iff flat Berry connection

I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$.
The first is the canonical or $H$-...

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### Some questions about the definition of Chern classes in Cheeger--Simons differential characters

In page 62 to 63 of the paper "Differential characters and geometric invariants" by Cheeger and Simons, they define, among other things, Chern classes taking values in differential ...

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### Precise definition of a linear total differential operator

In the works of A. M. Vinogradov on calculus on the infinite jet space, differential equations and "diffieties", a central notion is that of a $\mathcal C$-differential operator. If $\pi:Y\...

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### Are two homotopic principal bundles isomorphic?

Let $E_1 \to B$ and $E_2 \to B$ be two principal $G$-bundles, where $E_1$ and $E_2$ are two simply-connected manifolds and $G$ is a compact Lie group.
Suppose there exists a $G$-equivariant continuous ...

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### Minimal first Pontryagin class $p_1=1$?

From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...

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### Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$

In Whitehead tower of $BO$, there is a induced fiber sequence:
1.
$$
Z_2 \to B SO \to BO \overset{w_1}{\rightarrow} B Z_2
$$
How does this map $\overset{w_1}{\rightarrow}$ from $BO$ to $B Z_2$?
...

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### How exactly does the Kreck-Stolz description of elliptic homology match the one by Totaro?

In
Kreck, Matthias; Stolz, Stephan, $\mathbf H\mathbf P^2$-bundles and elliptic homology, Acta Math. 171, No. 2, 231-261 (1993). ZBL0851.55007.
the $n$th elliptic homology group of a space $X$ is ...

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### Orthogonal bundles with values in a line bundle vs. reductions of structure group to $O(n)$

I have already posted this question in math.stackexchange here, but didn't get any response, so I'm posting my question here as well.
Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\pi:...

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### Lifting action of torus to torus bundle

Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...

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### Integral sections of higher-order jet fields

I posted this topic on StackExchange, but it may suit this forum better.
Consider a bundle $(E,\pi, M)$ and let $k\in \mathbb N$. I am going to adopt the notations and conventions by Saunders.
...

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### Bianchi's identity in a principal bundle

Let us consider a principal bundle $P$, with a Lie-algebra-valued connection one-form $\omega\in\mathfrak{g}\otimes\Omega^1(P)$ and a Lie-algebra-valued curvature two-form $\Omega\in\mathfrak{g}\...

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### On the definition of smooth fiber bundle and smooth manifolds with boundary

On page 268 of Prof. John M Lee's book "Introduction to Smooth Manifolds" (second edition), it says if $E$, $M$ and $F$ are smooth manifolds with or without boundary, $\pi:E\to M$ is a ...

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### Twisted interval-bundles over a surface

I am trying to understand interval bundles over orientable surfaces. I know of course the basic examples: trivial interval bundles are just products. From what I understand, there is only one non-...

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### Lifting paths along group quotients relative to a base

Suppose you have a map of topological spaces $X\to S$, an $S$-group $G\to S$ (i.e. a group object in $\mathrm{Top}_{/S}$), an action of $G$ on $X$ relative to $S$ which is free and properly ...

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### Euler class of vertical tangent bundle of the surface bundle over circle

Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...

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### Obstruction to a cohomology class on total space being a pullback of a class on the base space is the restriction to the fiber

Let $\pi \colon E \to X$ be a fiber bundle with fiber $F$ and suppose that $\tilde H^i(F) = 0$ for $0 \leq i \leq k-1$.
Using the Leray-Serre spectral sequence, we get an exact sequence
$$
0 \to H^k(...

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### Frame bundle of $\mathbb{C}P^n$ as homogeneous space

I am reading "Dirac Operator in Riemannian Geometry" by T. Friedrich, where he writes that (the total space of) the frame bundle $R$ of the tangent space of $\mathbb{C}P^n$ is:
$$ R = SU(n+1)...

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### Reference request: Cut-and-project method gives rise to a fiber bundle over the torus

I apologize in advance for how vague this request is.
A few weeks ago, I came upon a paper that (if I recall correctly) proves that the hull of a cut-and-project tiling is a fiber bundle over a torus. ...

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### When is compactness of fiber components an open condition?

Consider a smooth map $f:M\rightarrow N$ between smooth manifolds.
Ehresmann's theorem states that if $f$ is a proper submersion, then it is locally trivializable; in particular, this implies that ...

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### Lifting a group action to a Banach bundle

I have been searching the literature for results on lifting a group action from the base space of a Banach bundle (Important note: NOT Banach VECTOR bundle). The setting I am interested in weaker than ...

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### What integral formula is being used here?

I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some ...

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### Gauge-natural lifts of principal connections

Let $P=(P,\pi,M,G)$ be a principal fibre bundle and $\omega$ a principal connection on it. If $\lambda:G\times S\rightarrow S$ is a smooth left action of $G$ on a manifold $S$, the associated fibre ...

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### The key step in Serre's method on higher homotopy groups

Let $n \geq 2$ and $X$ be a $(n-1)$-connected simplicial complex. This means that all of the lower homotopy groups $\pi_{k}(X) = 0$ for $k \leq n-1$. My goal is to compute the higher homotopy groups ...

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### Free $S^1$-action on compact homogeneous spaces

Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
If $r(G) > r(K)$ (...

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### Continuous choice of null directions for a family of bilinear forms

Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...

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### Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?

Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that
$\pi(g\cdot m)=...

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### Lifting smooth homotopies in smooth fiber bundles

I wish I had a reference for the following fact. Every smooth bundle on $M\times I$ is isomorphic to the pullback (via projection $M\times I\to M$) of a smooth bundle on $M$.

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### Riemannian geometry of Grassmannian bundles

The Grassmannian bundle of a vector bundle $E$ is a smooth manifold where each fiber over the base space is replaced by the Grassmannian (of specified rank) of the fiber. I am interested in defining a ...

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### Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle

Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\...

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### When does an analytic submanifold descend to the quotient?

Let $\pi : M \to B$ be a smooth principal bundle with group $G$, where $M$ is an analytic (Fréchet, in my case) manifold, $B$ is a smooth (Fréchet) manifold and $G$ is a smooth (Fréchet) Lie group. ...

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### Does fiber bundles admits good properties of covering spaces?

Let $X$ and $Y$ be non compact complex manifolds and $f:X\to Y$ be a holomorphic fiber bundle with fibers $F$ such that $f^*:\pi_1(X)\to\pi_1(Y)$ is injective and let for any $f_1,f_2\in F$ there ...

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### A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface

Let $N(n,k)$ denote the moduli space of stable vector bundles of rank $n$ and degree $k$ over a compact Riemann surface $X$, and let $N_0(n,k)$ denote the moduli space where we fix rank $n$ and some ...

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### Pull-back of factor of automorphy

Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\...

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### Fiber bundle orientability vs manifold orientability

This question seems like a pretty straightforward generalization of a result from vector bundles but its been on MSE for over a week with no answers so I'm reposting https://math.stackexchange.com/...

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### Projective bundle is stable under twisting by a line bundle [closed]

I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)≅P(E⊗L)$".
The statement can also be seen on the ...

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### Cohomology ring of mapping torus

A mapping torus, $M \rtimes_\varphi
S^1$, is a fiber bundle over $S^1$ with fiber $M$, where $\varphi$ is an element of mapping class group of $M$, describing the twist around $S^1$.
For $M=S^1\times ...

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### Stiefel Whitney number of a fiber bundle

I was going through this paper, and the author rights the following
The Stiefel-Whitney class of $E$ is given by $$w(E)=(1+\alpha)^{2m+1}\left\{(1+c)^{2n+1}+u_1(1+c)^{2n}+\dots+u_{2n}(1+c)+u_{2n+1}\...

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### Linear $S^{2k}$-bundles over $S^{4k}$

By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\...

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### Patching up two trivial fibre bundles induces homology equivalence

I was wondering to ask this question may be it's a silly one. I could not prove or disprove it.
Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap ...

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### Torus bundles and compact solvmanifolds

I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...

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### Non existence of preferred Horizontal subspace on a bundle [closed]

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=S^1 \times S^1$ and $B=S^1$. Can I follow the identity element of the group over a curve at the base. How ...

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### Isometries of the complex projective space for the Fubini Study metric

$\DeclareMathOperator\SU{SU}$I am trying to understand a geometric proof in our mathematical quantum mechanics lecture regarding Wigner's theorem in finite dimensions. We have already shown that it ...

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### What conditions are sufficient for the Leray-Hirsch theorem to be a Künneth formula?

This was originally posted on MSE, and since it didn't receive much attention, I'll try here. Let me know if this is not the appropriate place.
Given a fiber bundle $F \to E \to B$ over a paracompact ...

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### $ \mathbb{R}P^n $ bundles over the circle

Is every $ \mathbb{R}P^{2n} $ bundle over the circle trivial?
Are there exactly two $ \mathbb{R}P^{2n+1} $ bundles over the circle?
This is a cross-post of (part of) my MSE question
https://math....

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### Combinatorial fiber bundles

Triangulations (as simplicial complexes) and bi-stellar flips are a combinatorial analogue of (piece-wise linear) topological manifolds. I'm looking for a similar combinatorial analogue for fiber ...

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### Classification of functorial smooth vector fiber bundles

Let $\mathrm{Bundle}$ be the category whose objects are smooth vector fiber bundles over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).
Let ...