Questions tagged [vector-bundles]

For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.

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uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$

In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
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Induced group action on tangent bundle commutes with structure group?

I am trying to understand how the free and proper action of a discrete group $\Gamma$ on a manifold $X$ by automorphisms changes the structure group of the tangent bundle $\mathcal{T}_X$ of $X$. Let $\...
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Varieties with no vector bundles

What are some examples of algebraic varieties over the complex numbers with no (algebraic) vector bundles other than the trivial ones? The only example I can think of is $ \mathbb{A}^n_{\mathbb{C}} $ ...
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Is the Lagrangian canonical momentum the fiber derivative of the Lagrangian?

Consider a Lagrangian $L:TQ\to \mathbb{R}$, for some smooth manifold $Q$. As explained in this answer, one can define its fiber derivative $\mathbf FL:TQ\to T^* Q$ as $$\mathbf FL(v) \equiv \mathrm d(...
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Rank of a coherent sheaf using resolution by vector bundles

The rank of a coherent sheaf is defined in terms of the Hilbert polynomial (See Huybrechts-Lehn 1.2.2 or Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial). Now let $\mathcal{...
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Husemoller: homotopy of linear clutching map (proposition $4.5$, pag. $187$)

Background : I'm currently studying vector bundle through the book of [husemoller,"fibre bundles"] (https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller). The following question concerns a ...
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Does a covering by trivializations imply existence of a linearly compatible one, for a smooth $\mathbb{R}^k$ fiber bundle?

Let $\pi : E\rightarrow B$ be a surjective submersion between smooth submanifolds. Let $k=\mathrm{dim}\,\mathrm{ker}\,\pi_*$. We define a local trivialization of $\pi$ with fiber $\mathbb{R}^k$ to be ...
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The isomorphism $H^1(X;\mathbb Z_2) \rightarrow \operatorname{Hom}(\pi_1(X),\mathbb Z_2)$ and $w_1(E)$

On page $87$ of Hatcher's book Vector Bundles and K-Theory it states that, assuming $X$ is homotopy equivalent to a CW complex ($X$ is connected), there are isomorphisms $$H^1(X;\mathbb Z_2) \...
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A non-orientable vector bundle (rank $n$), for any collection of sections {$s_1, …,s_n$}, at least one subset of $E_b$ is zero? $s_i(b)=0$

Solved questions (see the comment below): For a vector bundle $E\rightarrow B$, if there are $n$ linearly independent sections {$s_1,…,s_n$} (i.e. trivial bundle). We take a linearly independent ...
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The $n$-th Stiefel–Whitney class $w_n \ne 0$ indicates that every section of the vector bundle (rank $n$) must vanish at some point

What I read: If $w_n \ne 0$, where $n$ is the rank of the vector bundle, then there cannot exist one everywhere linearly independent section of the vector bundle. The $w_n \ne 0$ indicates that every ...
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Does a non-trivial orientable vector bundle with rank $n$ have $n$ independent sections?

Why I know A bundle of rank $n$ is trivial iff it has $n$ linearly independent sections. We can write $B\times \mathbb R^n \rightarrow E(b,t_1,...,t_n)=\sum_i t_is_i(b)$ (where $t_i\in \mathbb R$) ...
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Splitting of short exact sequence of holomorphic vector bundle

I'm having trouble understanding the failure of splitting of holomorphic vector bundle. Following are my thoughts on this issue: In general, for a s.e.s. within the same category (by which I mean $E,F,...
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Why is for cotangent bundles $T^*X_1 \times T^*X_2 \simeq T^*(X_1 \times X_2)$

For smooth manifolds $X_1, X_2$ why can the cotangent bundle $T^*(X_1 \times X_2)$ be identified with $T^*(X_1) \times T^*(X_2)$, i.e. the product of the cotangent bundles. I feel like this should be ...
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Covariant and exterior covariant derivative of a bundle-valued $n$-form.

Let $E\to M$ be a vector bundle above a manifold $M$, with a connection $\nabla$ defined on the tangent bundle, and let $\nabla^{E}$ be a linear connection on $E$ and $\omega$ a $n$-form on $M$ with ...
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What is a symplectic structure on a smooth vector bundle

We just had the definition of a symplectic structure on a vector bundle in the lecture and I am having trouble understanding it Definition: Let $\pi : E \to M$ be a smooth vector bundle. Then a ...
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Why can extensions of a section of a vector bundle over a finite number of trivializing sets glue together?

This is motivated by the proof of Theorem 6.8 in Bott and Tu's Differential forms in algebraic topology. In the second paragraph there, they make a claim which can be generalized as follows: Suppose ...
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Chern forms and tensor products

Let $E\to X$ be a rank $r$ holomorphic vector bundle over a Kahler manifold and let $L\to X$ be a holomorphic line bundle. The following relation between Chern classes is (well) known: $$c_2(E\otimes ...
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The total space of the standard quaternionic Hopf fibration of an $S^3$ fiber bundle over $S^4.$

I am trying to understand the paper "Milnor's Construction of Exotic 7-Spheres" by Rachel McEnroe (link). Here's the abstract: I do not understand why the total space of this fiber bundle ...
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If the first Stiefel-Whitney class of a subset $\gamma$ of a vector bundle $E$ is $w_1(\gamma)\neq 0$, is $w_1(E)\neq 0$ as well?

Define: A vector bundle $E$ over a m-dimension base space $M$, $f: E\rightarrow M $ A non-contractible loop $L$ in the base space $M$ The vector bundle of the loop $L$ is $\gamma$ , $g: \gamma \...
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Understanding how the local trivializations were calculated.

I am trying to understand the paper "Milnor's Construction of Exotic 7-Spheres" by Rachel McEnroe (link). Here's the abstract: But when it comes to calculating the local trivialization of ...
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Understanding the transition map in the case of the proof that $S^3$ bundles over $S^4$

Here is the part of the paper of "Rachel Mcenroe" on Milnor's construction of Exotic 7-spheres: But I do not understand why the transition map is $\frac{1}{z}$ and it does not include any $\...
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Is $TM\oplus NM \cong M\times \mathbb{R}^n $ valid if the manifold $M$ is immersed in $\mathbb{R^n}$?

I understand that if a manifold $M$ is embedded in $\mathbb{R}^n$, $TM\oplus NM \cong M\times \mathbb{R}^n $. Is this equation still valid if $M$ is immersed in $\mathbb{R}^n$? For example, if we let ...
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Prove the first Stiefel-Whitney class of the line bundle (as a Mobius strip) over the base space $S^1$ is $w_1=1$

What I know: A trivial vector bundle $E$ has a vanished first Stiefel-Whitney class $w_1(E)=0$. If a vector bundle $E$ is non-orientable (as a vector bundle, not a manifold), $w_1(E)\neq 0$. The ...
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Prove the non-orientability of a vector bundle using the first Stiefel-Whitney class $w_1 \neq 0$

Question: How to compute the first Stiefel-Whitney class $w_1$ of the following vector bundle? Or Prove it is $w_1\neq 0$. The base space is a torus $T^2$ The fiber is locally $\mathbb{R}$ The line ...
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Why the fiber of the normal bundle of the Klein bottle is $\mathbb{R}^2$ instead of $\mathbb{R}$?

What I know: If M is an m-dimensional manifold embedded in $\mathbb{R}^{{m+k}}$, the normal bundle $NM$ is with the typical fiber $\mathbb{R}^k$. My Question: When I think about a torus ($T$) or a ...
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Normal bundle of a non-orientable manifold

What I know: A tangent bundle $TM$ of a manifold $M$ is orientable if and only if the manifold $M$ is orientable. If the first Stiefel-Whitney class $w_1(E) \neq 0$, the vector bundle $E$ is non-...
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The first Stiefel-Whitney class of a trivial line bundle $E=M \times \mathbb{R}$

If the base space $M$ is non-orientable, is the trivial line bundle $E=M \times \mathbb{R}$ also non-orientable? i.e. $w_1(E) \neq 0$. If so, how could it be proved? Could we use $w_k(\xi\times\eta)=\...
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Hermitian metrics on the Associated Vector Bundle of a Principal $U(1)$-bundle

If a complex line bundle $L$ over some manifold $M$ has a Hermitian metric, then $M$ has a frame bundle, which is a principal $U(1)$-bundle over $M$. Now reverse this, if we have a principal $U(1)$-...
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correspondence between local section and local trivalization for vector bundle and principle $G$-bundle

For the principle G-bundle $\pi :P\to M$, given a local section $s:U\to P$, it correspond to a local trivialization $$U\times G \to \pi^{-1}(U)\\(x,g) \mapsto s(x)\cdot g$$ where $s(x)\cdot g$ is the ...
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How does the Borel-Weil theorem works? What do we use it for, how does the application look generally?

Borel-Weil from Sepanski´s book "Compact Lie Groups": My intuition for Borel-Weil: We have a compact Lie group G. And we want to “describe it nicely” - we want to find its representation to ...
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Isomorphism in the space of section of a trivial vector bundle

In the answer of this post Fundamental result on the projective tensor product of sections of a vector bundle we have $\Gamma(M, V\times M) \cong C^\infty(M) \otimes V$ Where $\Gamma(M, V\times M)$ ...
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Adjoint of a end-valued complex differential form

Let $E\to X$ be a holomorphic Hermitian vector bundle over a complex manifold. Let $\xi\in \Omega^1(X,\operatorname{End}(E))$ be an end-valued form. We define its adjoint $\xi^*$ by the identity $$h(\...
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Splitting Lemma for Vector Bundles

I am asked to solve the following exercise. Let $E = E[M; \pi, \mathbb{R}^n]$,$F = F[M; \pi, \mathbb{R}^m]$, $H = H[M; \pi, \mathbb{R}^k]$ be three smooth vector bundles over $M$ of finite rank $n,m,k ...
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Let $(E,B,\pi)$ be vector bundle and $W \subseteq E$. Prove $W = \pi^{-1}(W \cap B)$.

I am having trouble proving a statement for my differential geometry class. The problem is as follows: Let $(E,B,\pi)$ be a vector bundle and let $\Psi : W \longrightarrow W' \times F'$ be a vector ...
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Isomorphism in the space of section of the external tensor bundle

Definition Let $E_{1} \xrightarrow{\pi_{1}} M_{1}$, $E_{2} \xrightarrow{\pi_{2}} M_{2}$ be two vector bundles over $M_{1}$ and $M_{2}$ with fibers $V_{1}$, $V_{2}$ respectively. The exterior tensor ...
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The module of section of a vector bundle and the module of section of the pullbackbundle are isomorphic

Let $\pi: E \rightarrow M$ be a vector bundle and let $f: N \rightarrow M $ be a continuous map. Let $f^{*} E$ be pullback bundle Let $\Gamma(E)$ be the module of section of the vector bundle $E$ and ...
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Proof that the module of section of a vector bundle and the module of section of the pullbackbundle are isomorphic.

I put this question in math.stackexchange but since I had no answer Let $\pi: E \rightarrow B$ be a fiber bundle with abstract fiber $F$ and let $f: B^{\prime} \rightarrow B$ be a continuous map. ...
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Confused on Projective Bundle

I am working through Atiyah (https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf) and am confused on the projective bundle (pg. 44-45). Atiyah states that for a vector bundle $E$ over $X$, let $...
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3 votes
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Gysin -/ Shriek map of projectivized bundle

Consider the Euler sequence $$0\rightarrow \gamma \rightarrow \epsilon^4 \rightarrow Q \rightarrow 0$$ over $\mathbb{P}^3$. Take the projectivized bundle $\pi:P(Q)\rightarrow \mathbb{P}^3$ and ...
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Extension of a Partial Section of a Fibre Bundle

I have to prove the following fact. Let $\pi :E\to M$ be a vector bundle of rank $k$, over a $n$-manifold $M$. If $S\subseteq M$ is closed and $s\colon S\to E$ is a partial section of $\pi$ (i.e. $\pi(...
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Harder Narasimhan polygon

We know for vector bundles on smooth projective curve we have a HN filtration. Let $0=E_0 \leq E_1 ... \leq E_n=E$ be such a filtration of $E$. Then we can construct a convex polygon with lattice ...
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Stable vector bundles on an algebraic curve

Let $E$ be a stable bundle on a smooth curve of genus $g$. Assume $\chi(E)\leq 0$ or equivalently, $E$ has slope $\leq g-1$. Is there a line bundle $L$ of degree 0 such that $H^0(E\otimes L)=0$ ?
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Pushforward of a line bundle

Let $\pi: X \to Y$ be a map between reasonable schemes. Let $L$ be a locally free sheaf on $X$, i.e. a line bundle. It's not always the case that $\pi_* L$ is a vector bundle but sometimes it is. For ...
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Intuition of semistable vector bundles over a curve.

Consider a characteristic zero algebraically closed field $k$. Let $f(x)\in k[x]$ be a polynomial of degree $2g+2$ ($g\geq 2$) with no double roots and $f(0)\ne0$. Let $g(u)\in k[u]$ be the ...
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Slope-stability and subbundle of a line bundle.

I have 2 questions. Let $X$ be a smooth projective curve. Let $L$ be a line bundle. Why is $L$ slope-stable? What I need to show is that every non trivial subbundle (in the sense that it is a ...
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Calculate the induced covariant derivative on the pullback bundle $\pi^*\mathcal{E}$

Let $ \pi: \mathcal{E}= M \times E \rightarrow M $ be a trivial vector bundle (where $M$ is smooth and $E$ is a finite dimensional real vector space). Let $\nabla = d + \omega $ be a covariant ...
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Definition of an $\mathcal{F}$-twist of an algebra

This discussion comes from this paper. We are considering a connected reductive group $G/\mathbb{C}$ with a Borel subgroup $B\subset G$ and corresponding Lie algebras $\mathfrak{g},\mathfrak{b}$. Also ...
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Can pointwise quotients be lifted to a local quotient for a family of vector bundles?

Let $X$ be a smooth projective curve. Let $E_T$ be a family of vector bundles parameterised by scheme $T$ i.e a vector bundle over $X \times T$. Let for each $t \in T$ we have a surjection from $O_{X \...
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Projectivization of a bundle is invariant under tensoring with a line bundle

I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)\cong P(E\otimes L)$". By bundle you can as well ...
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5 votes
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Connections on Principal bundles & Covariant derivatives on Vector bundles

Nowadays I'm reading "Differential geometry" written by Taubes. I have some problems and I guess that there may be some typos or I must get something wrong. Suppose vector bundle $E$ is ...
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