In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. (Def: http://en.m.wikipedia.org/wiki/Fiber_bundle)

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Is there a generalization of the Quaternionic Hopf fibrations and its natural connection?

For the quaternionic Hopf fibrations, $S^3\rightarrow S^7\rightarrow S^4$, we have a natural BPST connection form. Do we have some generalization of it? For example, the 'natural' connection form on ...
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53 views

Pontryagin class of a wedge product of vector bundles.

Let $E\to M$ be a real vector bundle over a differentiable manifold $M$ and let $p_{1}(E)$ denote its first Pontryagin class. I would like to know if there is any formula allowing to write ...
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47 views

Möbius band as line bundle over $S^1$, starting from the cocycles

The professor asked us to construct a non-trivial line bundle over $S^1$ by giving an open cover of $S^1$ and the cocycles. My idea was to take as open cover $U_1:=S^1\setminus\{0\}$ and ...
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48 views

examples for fibration not fibre bundle

We can use path space to make a map into a fibration. Generally, is this construction of fibration a fiber bundle? Or can someone give me some examples of fibration not fiber bundle?
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If a line bundle admits a non-vanishing section then it is trivial

Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$. I have to prove that the line bundle above is trivial. Idea: ...
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58 views

A fiber product is a fiber bundle

Let $F,B$ be topological spaces. A fiber bundle $E$ over the basis $B$ with fiber $F$ is a topological space $E$ endowed with a continuous surjection $\pi:E\to B$ such that there exists an open cover ...
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23 views

Are vector bundles special cases of étale bundles?

Is it possible to define vector bundles as particular instances of étale bundles? An étale bundle is a bundle $p:E\rightarrow X$ which is a local homeomorphism (as in Maclane-Moerdijk): every $e\in ...
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Difference between types of connections [closed]

For my background, I am familiar with the basics of differential geometry, especially Riemannian geometry, and in some more advanced topics relevant to physics, especially general relativity. Lately ...
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58 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
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General Fiber in Positive Characteristic

It is well-known that a complex polynomial, considered as a function $f:\mathbb{C}^n\to\mathbb{C}$, is a fiber bundle over a cofinite set of "atypical values" which include the singular values of $f$. ...
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isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? ...
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Gap between “fibration” and “fiber bundle”.

There are fibrations $E \rightarrow B$ which are not fiber bundles. Example: $E = [0,1]^2 / \text{middle vertical line segment}$ and $B=[0,1]$. In this example, $E$ has the homotopy type of a total ...
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30 views

Can a fiber bundle with fiber R and structure group Z_2 be considered a vector bundle?

I have a principal $\mathbb{Z}_2$-bundle, then by Borel construction I get a fiber bundle with fiber $\mathbb{R}$ and structure group $\mathbb{Z}_2$. Can I say that this is a vector bundle?
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Classical differential operators with complex functions on Riemannian manifolds

I am having some trouble understanding how to use the classical operators ($\nabla, \operatorname{div}, \Delta$) with complex functions on a Riemannian manifold $(M, g)$. Consider the formula ...
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Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
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Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
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Is the space of smooth sections of a smooth bundle a Fréchet Manifold?

I'm not very prepared on these concepts and i'm wondering if there're some good references addressing this problem... My aim is to present the problem of linearization for the Euler-Lagrange operator ...
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Weil: Fibre Spaces in Algebraic Geometry

I have spent a decent amount of time searching for the notes for Weil's Fibre Spaces in Algebraic Geometry, written by A. Wallace, both in print and online. Does anyone have a file of it they'd be ...
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109 views

Fiber bundles with category morphisms as fibers

Given a total space $E = M \times F$ of a fiber bundle where $M$ is a smooth manifold and $F$ is the fiber. The fiber $F_x$ corresponding to the point $x \in M$ is the set of morphisms between objects ...
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43 views

Intersection form on $S^2 \tilde \times S^2$

The intersection form on the nontrivial $S^2$-bundle over $S^2$, denoted $S^2 \tilde \times S^2$, can be written as $\left[\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right]$ with ...
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Polynomials as Locally Isotrivial Covers

Let $k=\mathbb{A}^1$ be algebraically closed of arbitrary characteristic. I am interested in understanding when a polynomial $f:\mathbb{A}^n\to\mathbb{A}^1$ defines a locally isotrivial family over ...
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Confused with notations about Leray's theorem for singular cohomology

The following theorem is copied from Bott's book Differential Forms in Algebraic Topology in Page 192: Theorem 15,11 {Leray's theorem for singular cohomology with coefficients in a commutative ...
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1answer
51 views

Fiber bundle with null-homotopic fiber inclusion

It is an exercise from Hatcher (exercise 31, page 392): For a fiber bundle $F \to E \xrightarrow{p} B$ such that the inclusion $F \hookrightarrow E$ is homotopic to a constant map, show that the long ...
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$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
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Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
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57 views

Homeomorphism between $S^2$ and $CP^1$ via uniqueness of quotient

I am trying to show that $S^2$ and $\mathbb{C}P^1$ are homeomorphic making use of the following result - see e.g. Jack Lee Introduction to topological manifolds. Let $Y \xrightarrow{\pi_1} X_1 $, $Y ...
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1answer
19 views

Is a principal bundle automorphism locally given by a left action?

Let $G\hookrightarrow P \xrightarrow{\pi} M$ be a principal bundle, denote by $\cdot$ the right action of $G$ on $P$. Let $f:P\rightarrow P$ a bundle automorphism (i.e. $f$ is a diffeo, $f(p \cdot g) ...
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Definition of structure group associated with fiber bundle.

I was studying fiber bundle from Spanier book.In that book there is a definition of structure group associated with a fiber bundle.But I am not able to understand the definition properly.Could ...
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How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
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31 views

Connection form on a frame bundle

Let $(E(M),\pi,M)$ be the frame bundle over a manifold $M$ of rank $n$. Consider a covering of $M$ by open neighborhoods $U_{\alpha}$. Let $s_{i}$ and $t_{j}$ where $i,j\in {1,...,n}$ be frames of $M$ ...
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1answer
28 views

slice charts of subbundles

Definition. Let $ \pi :E \to M $ be a smooth vector bundle of rank $r$. A subbundle of rank $k$ is a disjoint union $E'$ of $k$-subspaces, one for each fiber $E_p$ , such that $E'$ is an embedded ...
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Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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Relation beween transition functions of a principal fiber bundle and its dual

What is the relation between transition functions of a principal fiber bundle and its dual? As an example, consider the transition map of the frame bundle on a manifold of dimension $n$ as the ...
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Certain principle bundle structure on $\mathbb{R}^{n}\setminus \{0\}$

Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a principle fibre bundle. Here $\mathbb{H}^{2}$ is the Poincare upper plane ...
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54 views

Quotient manifold theorem provides a fibration?

It's known that if $G$ is a Lie group and $H\subseteq G$ is a closed subgroup, then the quotient map $p\colon G\to G/H$ is in fact a principal $H$-bundle, which follows from the existence of local ...
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A tangent vector to the unit tangent bundle $T_1S^n$ at $(x,\xi)$ can be written $(X,Z)$ with $X \cdot x = 0$ and $Z \cdot \xi = 0$.

Let $T_1S^n$ be the unit tangent bundle, $(x,\xi) \in T_1S^n$. Why can a tangent vector in $T_{(x,\xi)}T_1S^n$ be written $(X,Z)$ with $X \cdot x = 0$ and $Z \cdot \xi = 0$? This is claimed in ...
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section of a fiber bundle

I heard in class that not every fiber bundle admits a section. I am not sure why this is true, you can always pick a point on a fiber and follow it through as you glue local trivializations then you ...
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2-form as a curvature form

Let $M$ be a manifold and let $F$ be a global-defined $2$-form over $M$. Are there any conditions of $M,F$ such that $F$ is a curvature form of a connection on some line bundle ?
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Cohomology of the Thom Space of a Vector Bundle

The Thom space $T(E)$ of a vector bundle $E \to B$ with metric is defined as $D(E)/S(E)$, where $D(E)$ denotes the disk bundle and $S(E)$ denotes the sphere bundle of $E \to B$. I've been trying to ...
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1answer
51 views

A morphism of principal bundles is an isomorphism.

Let $p_1:E_1\to B$ and $p_2:E_2\to B$ be two principal $G$-bundles. Let $f:E_1\to E_2$ be a principal $G$-bundle morphism. I want to show that $f$ is an isomorphism. I read somewhere that it is ...
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Classify sphere bundles over a sphere

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of ...
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The pullback bundle

Consider a smooth fibre bundle $P:E\to B$ over a base $B$ with fiber $F$, and a trivializing family $\{(\phi_a:P^{-1}(U_a)\tilde =U_a\times F\}_{a\in A}$. Let the clutch functions of this family be ...
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Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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A differentiable fiber bundle exists for any family of clutching functions.

Lemma: Let $F$ be a smooth manifold and let $\{U_a\}_{a\in A}$ be a covering of a manifold $B$, and let $\{g_{ab}\}_{a,b\in A}$ be a family of clutching functions. That is, $g_{ab}:U_a\cap U_b\to ...
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1answer
39 views

The inverse of a smooth bundle map is smooth.

Consider smooth fiber bundles $P_i:E_i\to B_i,\quad i=1,2\quad$ with fiber $F$. Let $\tilde f:E_1\to E_2$ be a smooth bundle map. That is a smooth map which preserves the fiber $F$. Let $f:B_1\to B_2$ ...
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69 views

Pull-back of a fibration along a homotopy equivalence

Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ and $g:B\rightarrow B'$ be homotopy inverses. Denote by ...
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What is an infinitesimal automorphism of a connection?

Let $P\to M$ be a principal bundle over a differentiable manifold $M$, with fibre $G$, equipped with a connection $H\subset TP$. I have heard the term "infinitesimal automorphism of $H$" but I haven't ...
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28 views

Holomorphic Hermitian metrics

Let $E\to M$ be a complex vector bundle. A hermitian metric $h$ on $E$ is a hermitian inner product on each fiber $E_{p},\, p\in M$. Suppose that $M$ is also a complex manifold and that $E$ is ...
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How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
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What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...