# Tagged Questions

88 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 ...
26 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) ...
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 ...
132 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 ...
133 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 ...
109 views

### tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
71 views

### Contractibility of the space of sections of a fiber bundle

Let $\pi: E \to M$ a fiber bundle and $\Gamma(M,E)$ the space of smooth sections of the bundle with topology induced by the Whitney topology on $C^{\infty}(M,E)$. Assume that each fiber is ...
57 views

### vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
51 views

### Expressing $\mathbb{R} P^3$ as a fibre bundle

This question came up in office hours with my differential topology prof and since then I've almost settled on an answer. The question was whether we could write $\mathbb{R} P^3$ as a fiber bundle ...
81 views

### Understanding the Concept of Monodromy; case of Lefschetz Fibrations.

My question is on the concept of monodromy around critical points in a Lefschetz fibration $p: M^4 \rightarrow S^2$ (and monodromy in general), where $M^4$ is a 4-manifold and $S^2$ is the 2-sphere. ...
84 views

### Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
44 views

### Maps of $G$-bundles

A vector bundle is a $GL(\mathbb{R}^m)$-bundle with fiber $\mathbb{R}^m$. A principal $G$-bundle is a $G$-bundle with fiber $G$ (where the "$G$" in "$G$-bundle" embeds into $\mathrm{Aut}\, [G]$ by ...
47 views

### how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
92 views

### Trivialisation of Moebius strip

I've just started studying Advanced Geometry and I'm in trouble with a (stupid) exercise. It's about finding a trivialisation of the Moebius strip (I'll refer to it as $E$) viewed as a fibre bundle ...
106 views

### How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
62 views

### Will the pullback of homotopic maps give rise to isomorphic fibre bundles?

I know it's certainly right for the case of vector bundles, but what about fibre bundles?
44 views

### Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
69 views

### What's wrong with my argument to compute the vector bundles on $S^1$?

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don't know where did I make a mistake. So as circle is ...
119 views

### $\mathbb{S}^2$ as a fibre bundle

I know, by the Hopf fibration, that $\mathbb{S}^3$ is an $\mathbb{S}^1$-fibre bundle over $\mathbb{S}^2$. Can $\mathbb{S}^2$ be an $\mathbb{S}^1$-fibre bundle over some manifold $M$?
304 views

### When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
165 views

### Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
148 views

### Stiefel-Whitney classes and lifts of structure groups

Let $M$ be a compact, smooth Riemannian manifold with tangent bundle $TM$. I will not distinct between $TM$ and the associated $O(n)$-frame bundle. I believe the following statements are true, but if ...
111 views

### “killing homotopy groups” passage

I don't understand a passage in the famous article of milnor and kervaire: let $\xi : E \to S^n$ be a vector bundle (of rank $k$) and let $[\xi] \in \pi_{n-1}(SO_k)$ the map associated to $\xi$. Let ...
360 views

### Stokes for integration along the fiber

I want to use a version of Stokes theorem for integration along the fiber and I need some help in proving a general statement. Let $F$ be a $k$-manifold with boundary and let $E \to M$ be a smooth ...
872 views

### What does “locally trivial” do for us?

For the following we will work in the smooth category. (But examples in the topological category is also welcome.) The usual definition of a fibre bundle is Def A fibre bundle is the quadruple ...
439 views

### basic differential forms

Given a fiber bundle $f: E\rightarrow M$ with connected fibers we call the image $f^*(\Omega^k(M))\subset \Omega^k(E)$ the subspace of basic forms. Clearly, for any vertical vector field $X$ on $E$ we ...