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Questions tagged [fiber-bundles]

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For questions about fiber bundle, which is a space that is locally a product space, but globally may have a different topological structure.

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$\text{SO}(4)$ is homeomorphic to $\text{SO}(3)\times S^3$

Is there a reference for a proof that $\text{SO}(4)$ is homeomorphic to $\text{SO}(3)\times S^3$? Since $\text{SO}(4)$ acts transitively on $S^3$ with stabilizer $\text{SO}(3)$, we have a fiber bundle ...
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Induced representations built on sections of an associated vector bundle. Questions on notations

Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group: $$ {\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\...
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1answer
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Image of path under fiber transport of a right G-principal cover is G-equivariant

I have seen many different definitions of $G$-principal covers (which I assume are largely equivalent) so in this context, a right $G$-principal cover is a properly discontinuous action of $G$ on a ...
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1answer
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Various proofs: pairs of points in a circle = Möbius strip

It is well-known that the space of "pairs of points in a circle" (also called the symmetric square of $S^1$) can be identified with a Möbius strip. I don't know where the idea originates, ...
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35 views

A covering with connected tocal space and finite fibres

I have a covering map $p:X\to Y$ with $X$ connected and the fibres are finite. What could I conclude? Could I conclude that $Y$ is connected (or only locally)? Could I conclude that the fibers have ...
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1answer
29 views

Trivialising homeomorphisms for a principal $G$-bundle as $G$-space morphisms

Let $P$ by a principal $G$-fibre bundle over a locally-compact Hausdorff space $X$. Denote by $$ h: U \times G \to P|_{U} $$ a trivialising homeomorphism for a trivialising open set $U \subseteq X$. ...
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Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected?

Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected? What if we substitute $\mathbb{R}^n$ and $\mathbb{R}^m$ with open neighborhoods of ...
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32 views

Last step in Voisin's proof of Ehresmann's lemma

I'm reading Voisin's proof of Ehresmann's lemma, given in the book Hodge Theory and Complex Algebraic Geometry I as theorem 9.3; the proof is essentially reproduced in this answer. To summarize, there'...
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Finding a natural homomorphism between two relative cohomology groups

I'm trying to better understand how relative homology/cohomology works in order to make use of them for a specific example in mind. Here I'll be working with cohomology over the integers. Let $A \...
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If the bundle of orthonormal frames has a continuos/smooth global section will the bundle of spin frames also have one?

Let $(M,g)$ be a semi-Riemannian manifold with metric of signature $(p,q)$. I believe the signature of the metric is not relevant for this discussion so I leave it arbitrary (corrections to this are ...
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Existence of fibre metric compatible connections on vector bundle

Given a fibre metric $h$ over a vector bundle $E$, how can I show there always exist connections on $E$ such that it is compatible with $h$ in the sense that given any two smooth section $\sigma,\psi$,...
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129 views

Compactness about fiber bundle

I am working on a problem (problem 10.19 (d)) in John M. Lee's Introduction to Smooth Manifold. Assume that $\pi$: $E$ $\rightarrow$ $M$ is a fiber bundle with model fiber $F$, I need to prove if $E$ ...
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Dot product of functions on cosets

Let the measures of locally compact groups $\,K < G\,$ be $\, dk\,$ and $\, dg\,$, correspondingly. For a Hilbert space $\mathbb{V}$ equipped with a dot product $\,\langle~,~\rangle\,$, introduce ...
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Coproduct in the category of differential bundles

Just for clarity I briefly state the definition I am using. A differential bundle is a differentiable submersion $\pi: E\to B$, where $E$ and $B$ are differential manifolds. Observe that I am not ...
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22 views

Applying the Leray-Hirsch theorem on certain manifolds

Let $M$ be a connected closed orientable manifold with Euler characteristic $0$. Let $TM$ be the tangent bundle over $M$ and $SM$ be the induced sphere bundle with fiber $S^{n-1}$. Looking at page 432 ...
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Extension of a principal bundle

Let $G$ be a Lie group and $M$ a smooth manifold. The universal cover $\tilde M$ is a principal $\pi_1(M)$-bundle over $M$. Question: Why does any homomorphism $\phi:\pi_1(M)\to G$ induces a principal ...
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Integration along fibers

The following statements are from the book heat kernel and dirac operator chapter 1. " Let $\pi : M \rightarrow B $ be a fiber bundle with n-dimensional fiber, such that both M and B are ...
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1answer
50 views

Well definedness of the correspondence between connections on $B$ and sections of $J^1B$

Let $M$ be a smooth manifold and $\pi\colon B\to M$ a fiber bundle on $M$. A (Ehresmann) connection on $B$ is a subbundle $H$ of $TB$ such that $TB=H\oplus V$, where $V$ is the bundle of vertical ...
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How does BDiff$(F)$ classify smooth $F$-bundles and why is it topologized with the Whitney $C^\infty$-topology?

We consider a smooth fiber bundle $p:E \rightarrow B$ with fiber $F$. I want to understand how such a fiber bundle is classified by a map $B\rightarrow $BDiff$(F)$. Namely, we should have a bijection $...
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Isometries of fiber bundles

Let $F\to S\overset{\pi}{\to} B$ a fiber bundle or Riemannian manifolds with totally geodesic fibers (i.e. $F_x\cong\pi^{-1}(x)\subset S$ is a totally geodesic submanifold for each $x\in B$). Question:...
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What does it mean geometrically that $H^*(B\mathbb{Z}_2) = \mathbb{Z}[a]$?

I found cited in a physics paper that $H^*(B\mathbb{Z}_2) = \mathbb{Z}_2[a]$ and I wanted to know what geometric shapes this corresponded to. Notice the answer is a polynomial ring. Here $\mathbb{Z}...
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Bijection between quotient space of principal smooth and topological bundle.

I was reading the notes on bundles "Differential Topology of Fiber Bundles" by Karl-Hermann Neeb and on the final pages (138-139) "Homotopy theory of bundles" the author author ...
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Lie-Algebra Bundle and Wigner-Inönü Contraction

Here it is stated, that for a Lie-Algebra Bundle the isomorphism-class of the Lie-Algebra is locally constant, if the Lie-Algebra is rigid. Especially it is stated, that semisimple Lie-Algebras are ...
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About uniqueness of lifted paths in the phath lifting theorem in fiber bundles

Im trying to understand the path lifting theorem (in the context of locally trivial fiber bundles presented in the book "Manifolds, Tensor Analysis and Applications", by Jerrold E. Marsden ...
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1answer
162 views

Realising the non-trivial orientable $S^2$-bundle over $T^2$ as a quotient of $S^2\times T^2$

Using the fact that $\operatorname{Diff}^+(S^2)$ deformation retracts onto $SO(3)$, one can show that over any connected, closed, smooth surface, there are two orientable $S^2$-bundles, the trivial ...
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Fibration of $\mathbb{P}\left( \mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus n}\right)$ by global sections

Consider the bundle $\mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2} \to \mathbb{CP}^1$. The collection $A:=\left\{ \left( \bar{b}- \bar{a}\zeta, a +b\zeta \right) \mid (a,b) \in \mathbb{C}^2 \...
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1answer
52 views

Do the fiber bundles over an Abelian category form an Abelian category? [closed]

Assume I have an Abelian category $C$. Is the category of fiber bundles, where the fibers are objects of $C$ Abelian?
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1answer
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A subbundle of $(0,1) \times \mathbb{R}^m$ over $(0,1)$ is trivial if derivatives of all sections are in it

Let $K \subseteq (0,1) \times \mathbb{R}^m$ be a subbundle of rank $k$ of the trivial bundle $(0,1) \times \mathbb{R}^m$ over $(0,1)$, that is, $K$ is a differentiable submanifold of $(0,1) \times \...
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1answer
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Surface bundle with flat connection and monodromy representation

I saw the following statement: An $S_g$-bundle $p: E\to B$ admits a flat connection if and only if the monodromy representation $\rho:\pi_1(B)\to\text{Mod}(S_g)$ lifts to $\rho':\pi_1(B)\to\text{Diff}...
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Principal $G$-bundle maps and sections of Associated Bundles

These days, I am making my way towards the classification result about homotopy classes of maps from a CW-complex $Y$ to $BG$ and isomorphism classes of principal $G$-bundles over $Y$, where $G$ is a ...
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canonically induces structure on Poincaré dual space

Let Poincaré dual $PD(a)$ be a smooth, possibly non-orientable surface which represents a class in $H_2(M^3,Z/2)$ Poincaré dual to $a ∈ H^1(M^3, Z/2)$. What are the logic dependences of the ...
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Recover a vector bundle by (some) restrictions

Problem: Consider a smooth, projective variety $X$ in $\mathbb{P}^n$, consider a vector bundle $E\xrightarrow{\pi}X$ os rank $r$. Suppose that $\mathbb{P}^k$ is a subvariety of $X$, and we know that ...
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Metric tensor of a space as an induced metric from embedding in complex space, and dilations thereof.

Apologies, I know the title's a mouth-full. We can consider the standard unit three-sphere $S^{3}$ as being embedded in $\mathbb{C}^{2}$ with coordinates: $$\Phi=\begin{array}{c} \phi^{1}+i\phi^{2}\\ \...
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35 views

Basis of vertical vector fields in a fiber bundle from generators of the Lie algebra of the structural group

Let $E$ be a fiber bundle, $B$ its base manifold, $F$ its fiber and and $G$ its structural group of dimension $n$ greater than the dimension $r$ of $F$. We note $z$ the points of $E$ and $x$ the ...
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Normal bundle and determinant line bundle to the submanifold: Induced structures

Can someone explain step by step why this is true? (intuition) The normal bundle to the submanifold $PD(A) ≡ N_2 ⊂ M_3$ for oriented $M_3$ can be realized as determinant line bundle $\det TN_2$. The ...
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2answers
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Lifting a map to the total space of a circle bundle

Let $\pi:P \to M $ be a smooth circle bundle, so $S^1$ is the fibre, $f:N \to M$ a smooth map. I would like to know what are the necessary and sufficient conditions for $f$ to lift to a map $g:N \to P$...
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1answer
51 views

Computing the euler number of an oriented disc bundle over a sphere

Suppose $E\to S^2$ is an (smooth) oriented disk bundle. Let $D_1$ and $D_2$ denote the upper and lower hemispheres of $S^2$, respectively. Then $E|_{D_1}$ and $E|_{D_2}$ are trivial, so they are both ...
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1answer
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Obstruction to a Spin structure on a bundle ξ, and ξ ⊕ $n$ det ξ

In Ref, it says that: The obstruction to putting a Spin structure on a bundle $ξ (= Rn → E → B)$ is $w_2(ξ) \in H^2(B;Z/2Z)$. Pin± structures is that Pin− structures on ξ correspond to Spin ...
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1answer
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Decomposing a normal bundle

I'm reading a paper on symplectic geometry. I reached to a point where the author uses normal bundles ( which I know the definition but I've never work with). He actually decomposes a given normal ...
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G torsor v.s. Principal G-bundle

G torsor v.s. Principal G-bundle Are G torsors always Principal G-bundles? How are G torsors related to Principal G-bundles? Is that G torsor primarily for algebraic geometry? While Principal G-...
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“Horizontal” Foliation on a fiber bundle

Before I ask my question, I would like to build up the setting. Let $\pi:E \rightarrow M$ be a smooth fiber bundle. Since $\pi$ is a smooth submersion, there is an induced foliation $\mathcal{F}(\pi)...
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Reference about integration along the fiber and bundles

I am reading $\textit{Differential Forms in Algebraic Topology}$ by Bott. This book is superb but I would like to find other references (maybe more recent) that concern, in particular, topics about ...
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How to picture a horizontal lift using two different fibers?

Motivation: I am trying to define a connection that relates two different fiber bundles sharing a base space B, and think about parallel transport using both of them. Although I am still new to this ...
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Bott&Tu, Definition of the Euler class of a vector bundle

I have a question while reading chapter 11 of Bott&Tu, Differential Forms in Algebraic Topology. The book first defines the Euler class of an oriented sphere bundle (a fiber bundle with fiber $S^n$...
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2answers
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Non isomorphic principal $G$-bundles

Remark: throughout this post we work in the smooth category, so that all manifolds, bundles, maps etc are assumed to be smooth. An exercise asks me to show that there exists no principal $S^1$-bundle ...
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Fibre bundles and good pairs

Consider a fibre bundle $E\rightarrow B$. I was wondering when we can say that the pair $(E,B)$ is a good pair? By a good pair $(A,X)$ I mean two spaces $X\subset A$ such that $X$ is closed and is a ...
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1answer
55 views

Understanding a mistake I made regarding homomorphism between certain function spaces

Let $M$ be a closed $n$-dimensional smooth manifold. Let $X = \amalg_{p \in M} \text{Aut}(T_p M)$ be a fiber bundle over $M$ topologized in the natural way. It can be shown that the space of sections ...
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Components of vector field and pushforward through a section

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $p\in M$ and $u=\phi (p)$. Suppose we have $E$ has coordinates $(x,y)$ and we have the map $\psi :E\mapsto E$ given by $\...
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1answer
38 views

Local trivialization of a fiber bundle associated to a principal bundle

Let $(P,\pi,M,G)$ be a principal bundle, where $P$, $M$, $G$ are smooth manifolds (total space, base space and fiber=structure group respectively, so $G$ is both fiber and structure group) the ...
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1answer
49 views

Some questions about Bott & Tu - Differential Forms in Algebraic Topology, chapter 11.

I am reading chapter 11 of Bott & Tu - Differential forms in algebraic topology. And I have some questions about this section. 1: Let $\pi:E\to M$ be a sphere bundle with fiber $S^n$. For each $x\...

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