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1answer
6 views

A question about a fiber bundles

Let $M$ be a differential manifold, $TM$ its tangent bundle, $W(M)$ its frame bundle and $V(M)= TM-{0}$. Is the space $V(M)$ fibred over $W(M)$? With which structure group?
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14 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 ...
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0answers
21 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 ...
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1answer
36 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 ...
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1answer
24 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, ...
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1answer
65 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) ...
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3answers
59 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
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0answers
46 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 ...
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0answers
22 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
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1answer
40 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 ...
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1answer
17 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 $B$)?
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1answer
43 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
26 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 ...
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0answers
34 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
26 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
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1answer
39 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 ...
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1answer
24 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 ...
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1answer
68 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 ...
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1answer
45 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
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1answer
53 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
17 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
15 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
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1answer
29 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
29 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 ...
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0answers
19 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
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1answer
56 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
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1answer
61 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 ...
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1answer
28 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 ...
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1answer
17 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: $[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$ ...
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1answer
51 views

Understanding a statement related to a circle action on a principal bundle found in a paper

I am trying to understand a statement in the paper http://iopscience.iop.org/0951-7715/3/3/012/pdf/0951-7715_3_3_012.pdf I give details below so it should not be necessary to look at the paper. ...
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0answers
30 views

Relation between principal bundle automorphisms and maps in $\Lambda^0(M,Ad(P))$

Let $ G \hookrightarrow P \xrightarrow{ \pi } M $ be a principal $ G $-bundle over $M$. Denote by $\mathrm{Ad} (P) = P \times _{ \mathrm{Ad} } G $ the non-linear adjoint bundle. It seems to me that ...
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2answers
146 views

Why is the constant relating the chern class and curvature form always $2\pi i$?

I'm reading Milnor's book on Characteristic Classes. In Appendix C, Milnor shows the invariant polynomial of the curvature form and the Chern class differ by powers of $2\pi i$. He first shows that ...
4
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1answer
51 views

Characteristic classes for quaternionic bundles

In a nutshell, the derivation of the Stiefel-Whitney and Chern classes for a (let's say) closed space $X$ is as follows: For $k = {\mathbb{R}}$ or $\mathbb{C}$, any $k$-line bundle $\xi \to X$ is the ...
4
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1answer
116 views

Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same ...
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0answers
18 views

Equivalence between pullback connections of smoothly homotopic maps

Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we ...
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0answers
25 views

Induced measurable subbundle

Let $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ measurable and suppose that the application $$ \displaystyle{\begin{array}{rccl} h:&Z&\longrightarrow& ...
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0answers
52 views

A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
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1answer
202 views

Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
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0answers
42 views

lifting injective maps to injective maps in principal bundles

Let $i \, :Y \hookrightarrow X$ be an inclusion of (nice) topological spaces, and suppose that the induced map $\pi_1(Y) \to \pi_1(X)$ is injective. Then every lifting of $i$: $$ \tilde Y \to \tilde ...
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0answers
64 views

extending maps from spaces to their whitehead towers

Let $f \,: X \to Y$ be a map between connected spaces. Let: $$ X^{(k)} \to \ldots \to X^{(0)} \approx X $$ and $$ Y^{(k)} \to \ldots \to Y^{(0)} \approx Y $$ be whitehead towers for $X$ and $Y$. What ...
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1answer
69 views

example diagram of pullbacks and fiber products

I am going through Category Theory for Scientists. I am on section 2.5.1 Pullbacks. I am having trouble visualizing a pullback. Earlier in the book the author gives a nice diagram of an example of ...
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0answers
36 views

Are 4-dimensional mapping tori always spin?

We know that all compact orientable manifolds of dimension 3 are spin. In 4 dimensions, $CP^2$ is not spin. I would like to ask if all 4-dimensional compact orientable mapping tori are spin?
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1answer
53 views

“Bundle of metrics” on a principal bundle?

I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a ...
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0answers
28 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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0answers
95 views

measurable function induces a measurable bundle

Greetings I am preparing a work on bundles and I found this statement Let $V$ a topological vector space with $\dim V=n$ and $(E,\pi,M)$ a vector bundle continuous over $M$ (compact space). If ...
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1answer
19 views

generation of sub bundle

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for ...
4
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1answer
140 views

Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space ...
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3answers
140 views

let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...