The fibre-bundles tag has no wiki summary.
4
votes
1answer
99 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 ...
7
votes
1answer
268 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
161 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
votes
1answer
393 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.
...
1
vote
1answer
120 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
votes
0answers
102 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
121 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
vote
1answer
127 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. ...
6
votes
1answer
245 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
votes
1answer
474 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
votes
1answer
225 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
votes
1answer
117 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
66 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
votes
1answer
192 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
votes
2answers
237 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
139 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
votes
3answers
712 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
votes
1answer
131 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 ...
6
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0answers
175 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
votes
1answer
183 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 ...
1
vote
4answers
313 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
votes
4answers
816 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
votes
1answer
462 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 ...
0
votes
1answer
284 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
votes
1answer
359 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 ...
1
vote
2answers
1k 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
votes
1answer
505 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
vote
1answer
216 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
votes
0answers
198 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 ...
0
votes
0answers
225 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 ...
8
votes
2answers
431 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
285 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 ...
2
votes
2answers
430 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 ...
11
votes
4answers
959 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 ...
4
votes
0answers
153 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 ...
13
votes
1answer
963 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 ...
4
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0answers
351 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
votes
1answer
252 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 ...
5
votes
0answers
125 views
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
vote
3answers
497 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 ...
4
votes
2answers
314 views
Name for bundle of algebraic varieties over a smooth manifold
Consider a smooth manifold $M$ and a bundle $\pi\colon E\to M$ over it, where each fibre of $E$ is an algebraic variety. Is there a special name for this kind of bundle? The idea I have is that ...
7
votes
2answers
624 views
Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles
Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle.
Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial ...
0
votes
1answer
452 views
Is a fibre bundle over a vector bundle trivializable on each fibre?
Let $\pi:E\to M$ be a vector bundle over a closed smooth manifold and supose $\Pi:F\to E$ is a fibre bundle over the total space of $\pi$. I'd like to know if, restricted to $E_p$, the second bundle ...
9
votes
4answers
541 views
$S^n \to S^m \to B$ bundle: possible?
Sphere bundles and bundles over spheres are everywhere and are excellent things to get one's hands dirty with.
(1a) But when can we have a bundle $S^n \to S^m \to B?$ It seems like requiring the ...
8
votes
1answer
335 views
A sphere bundle map
I think this may all be classical bundle-theory. But I'm trying to read some old papers on classifications of bundles and the following came up as questions I couldn't immediately answer:
Consider ...
7
votes
2answers
965 views
Surface bundles over a surface
What can be used to distinguish two $\Sigma_g$-bundles over $\Sigma_h$ up to
(1) homotopy?
(2) homeomorphism?
(3) fiberwise homeomorphism?
(4) bundle isomorphism?
And can these always be computed ...
4
votes
3answers
194 views
Homology of bundles over a triangulated base and $A_\infty$-algebras
Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H_{*}(p^{-1}(\sigma)) \simeq H_{*}(F)$ the obvious map and let $\mathcal{S}$ be ...
6
votes
2answers
409 views
Restrictions of Diffeomorphisms
Notation: Let$M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $Diff(M)$ the group of diffeomorphisms of $M$ and $Imb(S, M)$ the group of smooth imbeddings of $S$ into $M$.
A classical ...
6
votes
2answers
952 views
Totally geodesic surfaces in fibered 3-manifolds
Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface?
(Of course such manifolds exist if the 'Virtually Fibered ...
9
votes
1answer
421 views
Bundle-to-function correspondence
To a morphism of sets $f\colon E\to B$ with finite fibers, one may assign a function $$|f^{-1}|\colon B\to{\mathbb N}$$ sending an element $b\in B$ to the cardinality of the fiber $f^{-1}(b)$.
To a ...
