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1
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0answers
80 views

Lifting a quadratic system to a non vanishing vector field on $S^{3}$

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 ...
4
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2answers
179 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
votes
2answers
132 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
157 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
300 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
112 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
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1answer
320 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
176 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
425 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
votes
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
120 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
votes
0answers
149 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
157 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
259 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
503 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
410 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
120 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
70 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
197 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
268 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
votes
1answer
144 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
777 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
137 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
207 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
209 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
335 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
922 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
563 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
302 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
367 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
votes
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
votes
1answer
573 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
231 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
219 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
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0answers
251 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
452 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
votes
1answer
297 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
487 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
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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 ...
4
votes
1answer
196 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 ...
14
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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 ...
4
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0answers
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
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
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0answers
128 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
593 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
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2answers
318 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
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2answers
659 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
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1answer
477 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
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4answers
557 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
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1answer
343 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 ...