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1answer
49 views

Euler characteristic of a singular fiber

I am trying to understand Kodaira's classification of fibers. In the table at page 41 of Miranda's book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf there is given the Euler number of the ...
1
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0answers
17 views

Principal bundle, horizontal and vertical subspaces

Given a principal bundle $P(M,G)$, we can decompose $$T_pP=V_pP\oplus H_pP.$$ I don't understand why $$[X,Y]\in H_pP$$ if $X\in H_pP$, and $Y\in V_pP$. Thanks!
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0answers
28 views

When the associated bundle to a $U(1)$-bundle is an holomorphic line bundle?

Let $P$ be a principal $U(1)$ bundle over a complex manifold $M$, and let $\rho\colon U(1)\to Aut(\mathbb{C})$ be the representation of $U(1)$ on $\mathbb{C}$ given by the standard multiplication. My ...
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0answers
68 views

Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If ...
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0answers
6 views

Reduction of the structure group

Given a vector bundle $V$ of rank $r$ over a curve. What proprieties should $V$ satisfies in order to admet a reduction of the structur group to $O_r(k)$ (resp. $SL_r(k)$, $Sp(r,k)$) Could you give ...
5
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1answer
56 views

Is there a fibre bundle with fibre homeomorphic to $\mathbb R^k$ which cannot be given the structure of a vector bundle?

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity ...
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0answers
19 views

Metric, torsion free connections on principal bundles

Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant ...
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0answers
20 views

Space of invariant sections

I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...
2
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0answers
42 views

Curvature of a principal bundle and the exterior covariant derivative

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...
2
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1answer
23 views

Is restriction of a chart is a chart necessarey in the Definition of Fibre Bundle

This is the definition of Fibre Bundle from the notes James F Davis and Paul Kirk: I think the condition 3 is superfluous. Because if you have a chart over $U$ $\phi : U \times F \rightarrow ...
4
votes
1answer
53 views

Fibered product of Hopf Fibrations

I am wondering: what is the vector bundle associated to the Hopf-fibration $S^{3}\to S^{2}$? What is the fibered product of two Hopf-fibrations? Thanks.
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0answers
28 views

Fibered product of symplectic fiber bundles

Let $P_{1}\to B$ and $P_{2}\to B$ be smooth fiber bundles over the same base manifold $B$, and let us assume that the total spaces $(P_{1},\omega_{1})$ and $(P_{2},\omega_{2})$ are symplectic ...
1
vote
1answer
17 views

Morphism of vector bundles covering maps of the bases

Let $E_{1}\to B_{1}$ and $E_{2}\to B_{2}$ be vector bundles over differentiable manifolds $B_{1}$ and $B_{2}$. What means that $F\colon E_{1}\to E_{2}$ is a morphism of bundles covering a map $f\colon ...
1
vote
1answer
26 views

Enumerating fiber bundles with fiber and base first Eilenberg-Maclane spaces

How to enumerate fiber bundles (maybe, only as spaces, not bundles) with fiber $K(A, 1)$ and base $K(B, 1)$? It seems to be connected with enumerating short exact sequences of form $0 \to A \to ... ...
0
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1answer
35 views

Proof of $G\rightarrow G/H$ is a Principal H bundle

Let $G$ be a Lie group and let $H$ be a closed subgroup (not necessarily normal). Then $G$ is a principal $H$-bundle over the (left) coset space $G/H$. I could proof that the fibers are all ...
1
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1answer
60 views

Inner product on the space of sections

Let $L\to M$ be a real line bundle over a manifold $M$, and let us denote by $\Gamma(L)$ its space of sections. I am trying to find a product in $\Gamma(L)$ to make it into an algebra. The naive ...
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0answers
27 views

Size of largest fiber

Suppose I have a map $f\colon X\to Y$ with finite fibers (where $X,Y$ are simply topological spaces for now, although maybe it's better to think of schemes). Is there a name for the quantity ...
3
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0answers
38 views

Locally a product equals fiber bundle?

I know the definition of a fiber bundle as a map $p:E \rightarrow B$ such that the preimage of any open set in $B$ is diffeomorphic to a product and fits in a commutative diagram with the projection ...
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0answers
16 views

given a specific vector bundle how to see whether the first Pontryagin class is zero

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
2
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0answers
21 views

Is there a fiber bundle approach to nonlinear oscillations?

I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles ...
0
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1answer
49 views

Relationship between $\Gamma (E)\otimes \Gamma (B) $ and $\Gamma(E\otimes B) $

I have a question that I've been thinking about for a while. So If $(E,\pi_E)$ and $(B,\pi_B)$ are some vector bundles over some mainfold $M$. What is the exact relationship between $$\Gamma ...
3
votes
1answer
23 views

how to see whether a bundle is trivial or not?

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
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0answers
25 views

principal bundle and the associated bundle

Let $G\leq O(n)$ be a subgroup of orthogonal group. Let $\xi$ be a principal $G$-bundle. Let $\xi[\mathbb{R}^n]$ be the associated vector bundle. If $\xi$ is not a trivial bundle, can we obtain that ...
0
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1answer
33 views

Projection of fiber bundle is a submersion

I'm just wondering about my proof for the following fact. I get the feeling it is almost trivial but I am still getting a feel for geometry and so it doesn't seem 'obvious' to me just yet. The ...
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0answers
20 views

Two different notions of covering homotopy?

In Steenrod's The Topology of Fibre Bundles, there are two different notions of covering homotopy. One is furnished in Theorem 11.3: Let $\mathcal B,\mathcal B'$ be two bundles having the same ...
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0answers
14 views

existence and uniqueness of projection maping

This is a question on Linear algebra by Greub. Let E be a vector space. Let $C(E)$ be the free vector space formed by E. Let $i_E:E\to C(E)$ be $i_E(x)=f_x$ where $f_x(y)=1$ if $y=x$ and $f_x(y)=0$ ...
3
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0answers
34 views

Terminology question : “half smooth, half topological” fibre bundle

First, I know (or I think I know...) the definition of fiber bundle, be it in the smooth or topological category. Here is my situation, which is kind of between the two: I have a smooth manifold $E$, ...
5
votes
2answers
366 views

Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors ...
2
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1answer
57 views

What is the topology of this quotient of $S^2 \times S^1$?

So suppose you take an $S^2$, then you put an $S^1$ fiber over it which degenerates by smoothly shrinking to a point at its poles. What is the topology of this space in more familiar terms (assuming ...
1
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1answer
44 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 $G$. ...
3
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1answer
102 views

line bundles over the circle

I read in various places that up to isomorphism there are only two line bundles ( 1-d vector bundles) over a circle, the trivial one and the mobius strip. On the other hand, when I make a mobius strip ...
1
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1answer
42 views

Fiber bundles and manifolds

Each vector bundle is an example of a fibre bundle with some extra structure. This extra structure provides the algebraic object consisting of all sections (continuos or smooth) of given bundle. When ...
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0answers
39 views

Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...
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1answer
117 views

What can we say about a vector bundle with trivial unit sphere bundle?

If you want to avoid the back story to this question, feel free to skip the paragraphs between the horizontal lines. Yesterday Georges Elencwajg asked me the following question (I'm paraphrasing): ...
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0answers
45 views

Trivializations of Möbius band

If $E=\mathbb{R}\times (-1,1)$ and $\forall (x,t), (y,t^{'}) \in \mathbb{R}\times (-1,1)$, $$(x,t)\backsimeq(y,t^{'})\Leftrightarrow \exists n\in \mathbb{Z} ; x=y+2n\pi , t=(-1)^{n} t^{'} $$ ...
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2answers
132 views

Spin structures, frame bundles, and trivializations over the 2-skeleton

While reading an introduction to Spin- and Spin$^{\operatorname{c}}$ structures (found here), I encountered the following definition: Let $E\to X$ be an oriented $\mathbb{R}^n$-bundle over a CW ...
0
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1answer
15 views

On the definition of transition maps of a principal bundle

. How are transition maps actting on the trivializations via some continuous left action $G \times F \to F$? $g_{\alpha \beta}(x)p . \Phi_{\alpha}(x,p) :=\Phi_{\beta}(x,p)$?
1
vote
1answer
16 views

transition maps of a principal bundle are smooh

A smooth principal fiber bundle is a smooth fiber bundle $\pi: E \to M$ together with a Lie group $G$ and a fiber preserving right action $E \times G \to E$ which restricts to each fiber freely and ...
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0answers
51 views

Reference request for studying on Fiber bundles

I am looking for some material (e.g. references, books, notes) to get started with Fiber bundles and vector bundles. Can someone help me? Thanks.
2
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1answer
49 views

Understanding the Homotopy Invariance of Fiber Bundle

I'm trying to understand the proof of Theorem 2.1 in "The Topology of Fiber Bundles" found online at http://math.stanford.edu/~ralph/fiber.pdf. What I don't understand is how do we actually define ...
0
votes
1answer
15 views

transition maps of a principal bundle are smooh

A smooth principal fiber bundle is a smooth fiber bundle $\pi: E \to M$ together with a Lie group $G$ and a fiber preserving right action $E \times G \to E$ which restricts to each fiber freely and ...
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3answers
320 views

“Drawable” Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use: When our base ...
2
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1answer
49 views

fibration of real projective space over complex projective space

From the fibration $$U(1)=S^1\to S^{2n+1}\to \mathbb{C}P^n,$$ can we quotient the action of $S^0$ and obtain a well-defined fibration $$ S^1/S^0=\mathbb{R}P^1\cong S^1\to ...
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2answers
53 views

Is it possible to build a fiber bundle whose fibers are different? (Or we should not call it a fiber bundle?)

Suppose there is a fiber bundle $E$. The base space is $M$ so that $\pi:E\rightarrow M$ is the projection. By the definition, the bundle has a typical fiber $F$ such that the local trivialization over ...
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0answers
47 views

Homogeneous polynomials and line bundles

In Huybrechts' Complex Geometry text, he makes the following claim: (Claim) "Any homogeneous polynomial $s \in \mathbb{C}[z_o,\dots, z_n]_k$ of degree $k$ defines a linear map ...
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1answer
93 views

Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities). the isomorphism classes of complex line bundles over $X$ the homotopy classes of ...
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0answers
79 views

Characteristic class integral: on what manifold does $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$ hold?

Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds? Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd ...
2
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1answer
46 views

This result holds in general or just for vector bundles?

If $(P,\pi, M)$ is a principal $G$-bundle, then given a left $G$-space $F$, using the $G$-product we can create a new bundle $(P_F, \pi_F, M)$ that is said to be associated to the first. Also, if we ...
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1answer
123 views

Leray-Hirsch Using Kunneth Formula from “Differential form in Algebraic Topology” by Bott and Tu

Kunneth Formula: Let M and F are manifolds. If M has a finite good cover then $H^n(M\times F)=\bigoplus _{p+q=n} H^p(M)\bigotimes H^q (F)$ Bott and tu says One can prove Leray-Hirsch theorem by the ...
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0answers
47 views

Group action on fibre homotopy group

Let $p : E \rightarrow B$ be a Serre fibration, with typical fibre $F \cong p^{-1}(b)=: F_b$ for each $b \in B$. I know that $\pi_1(F) \looparrowright \pi_k(F)$, but now I would like to find a way to ...