The fibre-bundles tag has no wiki summary.
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196 views
Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?
Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V ...
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0answers
51 views
discrete group action on Stiefel manifold
Let $S_3$ be the permutation group of order $3$. Let $V_2(\mathbb{R}^n)$ be the stiefel manifold of $2$-frames in $\mathbb{R}^n$. Let $S_3$ act on $V_2(\mathbb{R}^n)$ by
$$
(1,2)(u,v)=(-u,v-u),
$$
$$
...
2
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1answer
131 views
A Dold-Thom style construction of a cohomology class from a sphere bundle
Re-reading my comment to the question Pontryagin class of quaternionic line bundle I suddenly realized that I do not understand something crucial about it. For the purposes of that crucial thing let ...
12
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1answer
249 views
Anything between vector bundles and sphere bundles?
There are two extremities: on the "easy end" one has vector bundles which are classified by maps to the (more or less) well understood spaces like Grassmanians; on the "hard end" there are spherical ...
1
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1answer
137 views
fiber sequence of principal bundles
Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$.
Let $BG$ be the classifying space of $G$.
...
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0answers
190 views
Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$
Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
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2answers
192 views
The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle
What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?
3
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2answers
139 views
Differentiable structure on the Gauge group of a principal bundle?
I am currently reading this paper in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most ...
4
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1answer
223 views
A generalization of covering spaces to fiber bundles with totally path-disconnected fibers
There is a classical theorem about covering spaces and the actions of the fundamental group.
Theorem 1: Let $B$ be a non-empty locally path-connected and path-connected space. The category of ...
14
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3answers
313 views
Existence of sections of the evaluation map for the diffeomorphism group
Let $M$ be a closed connected oriented smooth manifold and $\mathrm{Diff}_{+}(M)$ the group of orientation preserving diffeomorphisms of $M$ endowed with the compact-open topology. Pick a base point ...
4
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1answer
116 views
connections on principal bundles over $S^1$
Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My ...
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1answer
347 views
Two questions about sphere bundles
I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and ...
2
votes
1answer
184 views
Is this sphere bundle over SL3/SO3 trivial?
The space $SL_3(\mathbb{R})/SO_3(\mathbb{R})$ can be though of as some kind of 5-(real)dimensional generalized upper half-space.
Fix any copy of $SO_2(\mathbb{R})$ sitting inside $SO_3(\mathbb{R})$, ...
3
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1answer
431 views
How was this Lie algebra found?
In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself.
...
2
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1answer
126 views
Fibre bundles of fibre genus greater than 1
What are concrete examples of fibre bundles with fibre genus $\geq 2$? I am trying to find examples I can use to work throught the following construction:
Suppose $X \to C$ is a fibre bundle with ...
3
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0answers
123 views
Orthonormal frame bundle orthogonal to a curve
This is a duplicate of this question on math.stackexchange, since I got there not a single answer.
Let $M$ be a $n$-dimensional smooth riemannian manifold and ...
2
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0answers
160 views
Global sections for torus fiber bundle
Let us consider the following situation: $\pi:X\rightarrow B$ is a locally trivial fibration between smooth manifolds, its fiber being a torus $T$. My question is two-fold:
1) what is the obstruction ...
1
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1answer
176 views
on two definitions of irreducible connection
I have seen two kinds of definitions of irreducible connections on fibre bundls: A connection is said to be irreducible if
the holonomy group is precisely $G$ and not a proper subgroup.
or
2. ...
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1answer
273 views
how to obtain a generalized Morse function out of a fiber bundle?
I already asked this question in MSE but did not get any answer/comment yet.
Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, ...
8
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1answer
523 views
Local trivializations of the non-trivial $SU(2)$-bundle over $S^5$
It is well known that $SU(3)$ is the unique, non-trivial, principal $SU(2)$- bundle over $S^5$. To my knowledge the way this is proven is by using the following fact:
If $G$ is a Lie group ...
0
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1answer
452 views
fiber bundle on an orbit of $\mathfrak{g}\oplus\mathfrak{g^*}$
Let $G$, be a Lie Group and $\mathfrak{g}$ be its Lie algebra ,i.e, $Lie(G)=\mathfrak{g}$. Let $\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^*}$. Here $X\in \mathfrak{g} $ and $F\in ...
3
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1answer
126 views
Which varieties of general type admit fibrations with non-general type fibres
Disclaimer. I don't know much about the things I'm asking. This is why my other question pencils on varieties of general type was a bit unclear. I believe the following question makes up for this.
...
2
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0answers
75 views
Non-clean fiber products
Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g.
When do fibre products of smooth manifolds ...
2
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1answer
203 views
Confusions over the definitions of universal bundle and characteristic class
In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition).
Does it make sense to speak of a universal F-G ...
2
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2answers
278 views
Homotopy Equivalences and Induced Correspondences between Fibre Bundles
Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...
3
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1answer
145 views
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 ...
8
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3answers
810 views
Serre Spectral Sequence of Representations
Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...
0
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1answer
143 views
When is a sheaf of groups (algebras, rings, modules) a group (algebra, ring, module)?
If $\pi:E\to M$ is a vector bundle then the set of sections $\Gamma(E)$ is naturally a vector space under fibrewise addition and scalar multiplication on the bundle $E$. This holds similarily for ...
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0answers
226 views
Flat morphisms whose fibers are affine spaces
Let $f:X \to Y$ be a flat morphism, such that each fiber is isomorphic to the affine space $\mathbb{A}^n$. Then is is true that $f$ is a Zariski affine bundle? If not, is it at least an ètale affine ...
3
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1answer
235 views
twisted bundle definition
I have often heard people talk about, say, "the" twisted $S^2$-bundle over $S^2$.
My question is, what do they mean by a twisted bundle? I know that in the above example any $S^2$-bundle over $S^2$ is ...
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4answers
343 views
Triviality of a differentiable sphere bundle
I have read in some lecture notes in the internet (without reference) the following result:
Let $E$ be a differentiable sphere bundle whose base $B$ has dimension $n\geq 2$ and whose fibers $F$ ...
3
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4answers
989 views
Nontrivial examples of non-trivial principal circle bundles
It is a well known fact that (isomorphism classes of) principal $S^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z})$.
There is always the ...
2
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1answer
611 views
Cross sections in bundles and principal G-bundles
A principal $G$-bundle has a cross section iff it is trivial (e.g. Husemoller's Fibre Bundles, 3rd ed., 8.3 in chapter 4).
A principal $G$-bundle is in particular a fiber bundle with fiber $G$.
My ...
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1answer
315 views
Cotetrad, spin connection and Dirac operator
Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) ...
9
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1answer
374 views
representatives of the group of homotopy 7-spheres
In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere ...
4
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3answers
2k views
Vector bundles vs principal $G$-bundles
It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres
$$F=\pi^{-1}(x), \ \ \ x\in B $$
over any $x\in B$, are ...
3
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1answer
611 views
The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base
Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. ...
1
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1answer
235 views
Intersection of subvector bundles
Suppose we have a smooth vector bundle $\pi: E \rightarrow B$ and two sub vector bundles $\pi_1: E_1 \rightarrow B_1$ and $\pi_2: E_2 \rightarrow B_2$ such that
the bases $B_1$ and $B_2$ are ...
3
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0answers
226 views
Fibred manifolds with boundary
A fibred manifold is a triple $(E,\pi,M)$ where $E$ and $M$ are manifolds and $\pi : E \rightarrow M$ is a surjective submersion. (Saunders, The Geometry of Jet bundles)
A special case of this is the ...
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265 views
On the universal pullback of fiber bundles
First suppose we have three smooth manifolds $M_1$, $M_2$ and $N$ with
smooth transversal maps $p_1: M_1 \rightarrow N$ and $p_2: M_2 \rightarrow N$
then its a well known fact that the categoric ...
9
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2answers
465 views
Classification of holomorphic disc bundles
I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber ...
3
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1answer
301 views
degeneration of quadric bundles
Suppose I have a smooth 2-dimensional quadric bundle $f:X\to S$ over a surface $S$. Suppose furthermore that the discriminant locus $\Delta \subset S$ is smooth. Can I immedately conclude that the ...
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2answers
509 views
Elementary transformations of ruled surfaces as maps of vector bundles
This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$.
All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as ...
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4answers
1k views
non-locally trivial A^n bundles
Let $f: X \to Y$ be a morphism of varieties such that its fibres are isomorphic to $\mathbb{A}^n$. Since the definition of a vector bundle stipulates that $f$ be locally the projection $U \times ...
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1answer
197 views
isomorphism of extensions by abelian varieties
Let $A$ and $G$ be abelian varieties over $\mathbb{C}$. An element $P$ of $\text{Ext}(A, G)$ is an exact sequence
$0 \to G \to P \to A \to 0$,
here one can give $P$ the structure of an abelian ...
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1answer
1k views
Which principlal bundles are locally trivial?
If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial.
In the wikipedia article on ...
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358 views
What is the spectrum of the commutative C*-algebra I have constructed here?
Let $B$ and $F$ be compact Hausdorff spaces.
Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$.
I think this induces a fiber ...
3
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1answer
253 views
Reference for monkeying with the topology of a mapping cylinder
In "Construction of Universal Bundles, II", Milnor has to replace the standard topology on the join with what he calls the "strong topology" which is the smallest topology such that certain maps are ...
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Joins and classifying spaces in the category of compactly generated spaces
In Milnor's Construction of Universal Bundles, II, he defines $E_nG$ by repeated
joins of $G$ with itself, but he has to use the `strong topology' on the join instead
of the everyday topology that ...
1
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3answers
601 views
Topology of maps between fibers of vector bundles
First of all sorry for the (possible) incorrect english. I don't know english very well.
I'm with a doubt about topology of maps between fibres of vector bundles.
Consider $E$ and $F$ vector bundles ...
