Questions tagged [fibre-bundles]
for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.
338
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Deriving the definition of vector bundle morphisms from Cartan geometry (a.k.a. why are they linear?)
I'm familiar with the definition of the category of vector bundles, but I'm trying to derive it from some first principles about general fiber bundles. My intuition is that vector bundles should be ...
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Is every covering space associated to a "principal" covering space? Can they be classified?
Every fiber bundle is associated to a principal bundle with the same structure group. A covering space is just a fiber bundle with a discrete fiber, and hence a structure group $G$ that is a (subgroup ...
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Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
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Co-index of a Space
I am going through this paper by Tanaka. But I got stuck at the Proposition 2.4 given below as well.
Here he doesnot provide any proof rather just refers to Theorem 6.6 of another paper. Again given ...
2
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1
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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}\...
CommunityBot
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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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Restrictions of diffeomorphisms
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Imb{Imb}$Notation: Let $M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $\Diff(M)$ the group of diffeomorphisms of $M$ and $\Imb(S, M)$...
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A relative version of Ehresmann's theorem
Edited: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.
Let $N\subset M$ be a pair ...
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Fiber-bundle : continuity of transition maps and inverse in general
Let $(E,\pi,B)$ be a locally trivial fibration, with fiber a topological space $F$, $\Phi_i$ and $\Phi_j$ two trivializations over $U_i$ and $U_j$. The transition map from $i$ to $j$ is the ...
CommunityBot
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Anomaly in QFT physics v.s. determinant line bundle
In a quantum field theory (QFT) lecture, a math-physics professor explains the anomaly in physics, say the non-invariance of the partition function of an anomalous theory under background field ...
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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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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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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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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
$$
\mathbb{C}P^1 \xrightarrow{\iota} \mathbb{...
- 18.8k
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Some questions about the definition of Chern classes in Cheeger--Simons differential characters
In page 62 to 63 of the paper "Differential characters and geometric invariants" by Cheeger and Simons, they define, among other things, Chern classes taking values in differential ...
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Precise definition of a linear total differential operator
In the works of A. M. Vinogradov on calculus on the infinite jet space, differential equations and "diffieties", a central notion is that of a $\mathcal C$-differential operator. If $\pi:Y\...
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Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$
In Whitehead tower of $BO$, there is a induced fiber sequence:
1.
$$
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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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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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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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 ...
- 205
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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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1
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How to classify von Neumann algebra bundles?
If we consider algebra bundles over X where the fiber is an algebra of bounded operators in a separable Hilbert space H over the complex numbers. I learn from "Isomorphism Classification of Operator ...
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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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Models for Eilenberg-MacLane space K(Z,3)
Denote by $U(H)$ and $PU(H)$ the unitary and projective unitary groups on an infinite-dimensional Hilbert space $H$. Recall that $U(H)$ is contractible by Kuiper's theorem and that $PU(H)$ is a $K(Z,2)...
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2
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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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1
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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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Critical points on a fiber bundle
Consider a (smooth) bundle $E\to B$, and a (smooth) function $f: E\to\mathbf{R}$ on the total space. Then it makes sense to talk about the derivatives of $f$ along the fibers. Let $C$ be the ...
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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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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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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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1
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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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When is the cohomology of a fiber bundle a tensor product?
Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...
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2
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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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1
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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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1
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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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3
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Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
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Looking for an introduction to orbifolds
Is there any source where the basic facts about orbifolds are written and proved in full detail?
I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more ...
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2
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Relative minimality for conic bundles
Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces.
The definition is ok: the fiber over ...
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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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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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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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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)=...
- 1,908
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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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0
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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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