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1answer
11 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
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1answer
13 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 ...
3
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0answers
43 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
41 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
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1answer
14 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 ...
7
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3answers
191 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 ...
1
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1answer
22 views

what is a fiber for function mapping?

I have read that a fiber is the pre image of a mapping.. does this mean that i can think of a fiber as a line that connects x to y where the line is a function? so for example with $f(x)=2x$ there ...
2
votes
1answer
41 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 ...
2
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2answers
44 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 ...
1
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0answers
28 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 ...
2
votes
1answer
65 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 ...
1
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0answers
74 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
votes
1answer
40 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 ...
3
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0answers
72 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 ...
0
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0answers
42 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 ...
3
votes
1answer
30 views

On the existence of some kind of “universal fibre bundle”

While attending an introductory course on the theory of (smooth) fibre bundles, an example I was given of (principal) bundles was that of topological coverings of a space with structure group the ...
2
votes
1answer
44 views

function of class C ^ 1 on manifolds

Let $M$ be a differentiable manifold with finite dimension $ m $. Let $ f:M\rightarrow M $ a function of class $C^1$. I have a doubt about what this implies (1) or (2): $x \in M \rightarrow D_xf ...
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0answers
16 views

vector bundle and convergence of operators

Let $ (E, B, π, V)$ a fiber bundle. Suppose that $x_n \rightarrow x$ with $(x_n), x\in B$ then by the local triviality condition $\phi(x_n,.):V \rightarrow E_{x_n}=\pi^{-1}(x_n)$ is an isomorphism for ...
1
vote
1answer
28 views

Is a virtual vector bundle the same as a vectorial bundle?

What is a virtual vector bundle? Is a virtual vector bundle the same as a vectorial bundle? The current entry in nLab states the following: "In one class of models for K-theory – generalized ...
3
votes
0answers
30 views

quaternion vector bundle and quaternion grassmannian

Let $\mathbb{H}$ be quaternion numbers. Let $G_n(\mathbb{H}^\infty)$ be the grassmannian of $n$-subspaces of $\mathbb{H}^\infty$. Then ...
0
votes
0answers
19 views

whitney class of fiber bundles over classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $\rho: \Sigma_k\to GL(k)$ be regular representation by permuting basis. Let $\rho': B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced ...
0
votes
0answers
23 views

classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...
0
votes
1answer
42 views

Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?

Let $\pi: E \to M$ a vector bundle over a smooth manifold $M$. $E$ is direct summand of $M\times\mathbb{R}^{d}$, if there exist a vector bundle morphisms $f:E\to M\times\mathbb{R}^{d}$ and ...
0
votes
1answer
26 views

cup product of stiefel-whitney class

Let $\xi$ be a vector bundle. Let $w(\xi)$ be the total Stiefel-whitney class. Let $\bar w$ be the dual Stiefel-whitney class. In John Milnor's Characteristic class book, page 40-41 Chap.4, ...
2
votes
1answer
72 views

How can I visualize principal bundles?

I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization $ϕ^{-1}_i:π(U_i)→U_i×G$ . For example $ϕ^{-1}_i:π(S^2)→S^2×U(1)$ (if that makes sense) ...
1
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3answers
69 views

does a commutative diagram implies pull-back?

Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaystyle\bar f>> E'\\ ...
2
votes
0answers
57 views

automorphisms of the torus bundle over the circle

Let us consider a $\mathbb{T}^2$-bundle $\xi$ over circle $S^1$. These bundles are completely determined by their monodromy matrix. A fiber-preserving homeomorphism of $\xi$ is called automorphism of ...
0
votes
0answers
32 views

Tangent bundle to 2-sphere isnt trivial as a vector bundle?

I read many quiestion about $TS^{2}\ncong S^{2}\times\mathbb{R}^{2}$ where the hint is use Hairy ball theorem and directly is done. My question is: how do I proof that $TS^{2}\ncong ...
1
vote
1answer
45 views

classifying space of permutation groups

Given a group $G$, how to compute the classifying space of $G$? In particular, how to compute the classifying space $B\Sigma_n$ for a permutation group $\Sigma_n$?
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0answers
11 views

The second cohomology of total space of the $\mathbb CP^1$ bundle

$X$ is a closed smooth surface with $L$ a complex line bundle on $X$. Consider the $\mathbb CP^1$-bundle $P(L\oplus 1)$, that is the projectivization of the sum of $L$ and the trivial line bundle on ...
0
votes
1answer
20 views

Invariant forms on principal bundles

Let $\pi:M \to B$ be a principal $G$-bundle and $\xi$ a invarint $k$-form on $M$. Does $k> dimG$ implies that $\xi$ is a basic form (pull back of a $k$-form on the base manifold $B$)?
1
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1answer
51 views

Orientable surface bundles over the circle and their structure group

I want to understand whether orientable surface bundles over the circle, i.e. with orientable total space, are always trivial, so I though I would revive an old post and ask for a few clarifications, ...
1
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1answer
32 views

Trivial nature of orientable fibre bundle with cylinder base space

Given a bundle whose base space is a "cylinder", i.e. $\mathbb{S}^1 \times \mathbb{B}^n$ where $\mathbb{B}^n$ is an $n$-ball, is orientability (of the total space) enough to ensure that the bundle is ...
0
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0answers
35 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
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0answers
27 views

homotopy group of the limit space

Let $V_k(\mathbb{R}^{n+k})$ be Stiefel manifold. Using $\pi_i(V_k(\mathbb{R}^{n+k}))=0$ for all $i\leq n-1$, how to obtain $\pi_i( V_k(\mathbb{R}^{\infty}))=0$ for all $i\in \mathbb{N}$? Can I just ...
1
vote
1answer
53 views

Construction of a covering space as a fibre bundle

In a direct proof of the equivalence of categories between the covering maps $p:(\hat X, \hat x) \rightarrow (X,x)$ of a topological space $(X,x)$ for sufficiently beautiful $X$ and the ...
0
votes
1answer
25 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
1
vote
1answer
71 views

Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
1
vote
1answer
51 views

Étalé space for sheaf of sections of a fiber bundle

Let $X$ be a topological space, $\pi:E\to X$ a fiber bundle over $X$ with fiber $F$ and structure group $G$. Let $\mathcal{F}$ denote the sheaf of continuous sections of the bundle. I probably want to ...
4
votes
1answer
67 views

map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
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0answers
42 views

Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
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0answers
42 views

Connecting homomorphism in the Gysin sequence

Let $j : SO(2n) \to SO(2n+1)$ be the standard subgroup embedding and let $Bj : BSO(2n) \to BSO(2n+1)$ be the induced fibration obtained by factoring the universal $SO(2n+1)$-bundle by the subgroup ...
1
vote
1answer
37 views

spin structure definition

Suppose we have a principal $SO(n)$-bundle $E$ over $B$, with projection map $p$. We say that it admits a spin structure if there is a prinicipal $spin(n)$-bundle $E'$ over B, with projection map ...
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0answers
35 views

Sections of endomorphisms of a vector bundle

The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments. The problem is the ...
0
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0answers
20 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
4
votes
1answer
68 views

Canonical connection on $CP^n$

I have heard something along the lines of "There is a canonical $U(1)$ connection on $CP^n$" and I am trying to understand what that means. First I suppose that the sentence refers to a line bundle ...
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0answers
17 views

A construction on principal bundles

In a paper the principal $Sp(1)$-bundle $P$ over $S^4$ is introduced as follows: let $Sp(1)\times Sp(1)\hookrightarrow Sp(2) \xrightarrow{\pi} S^4$ be the spin structure on $S^4 $. The principal ...
1
vote
1answer
66 views

Group of Automorphism of the Fiber of Fiber Bundle

Let us consider the Mobius Bundle. Can Someone Explain the part "There is no natural unique homeomorphism of $Y_x$with $Y$. However there are two such which differ by the map $g$ of $Y$ on itself ...
1
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1answer
29 views

Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
0
votes
1answer
19 views

Linear clutching function properties

I'm trying to prove proposition $4.8$ of Husemöller (pay 148) Here it is the text: Where the only hints are: $[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$ ...