For questions on vector bundles.

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3
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0answers
35 views

Definition of the $\Bbb C^*$-weight of a line bundle

I'm new to geometric invariant theory and am unsure about a definition. Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this ...
0
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1answer
53 views

Quotient of locally free sheaf is locally free?

If $0\rightarrow F\rightarrow G\rightarrow H \rightarrow 0$ is an extension of $\mathcal{O}$-modules with $F$ and $G$ locally free (each of constant finite rank, i.e. vector bundles), then is $H$ ...
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0answers
16 views

K-theory product trivial on $K^1$ due to stability?

Let $K^0$ be the topological complex vector bundle $K$-theory functor. On the one hand, we have that the product on $K^0(\Sigma X)$ vanishes by a Mayer-Vietoris argument. On the other hand, we define ...
0
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0answers
13 views

Vector bundle, nonexistence of Euclidean metric

Milnor-Stasheff "Characteristic classes" problem 2-C says: Any vector bundle over a paracompact base space can be given a Euclidean metric in other words, if $\pi : E \rightarrow B$ is a vector ...
1
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1answer
102 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
3
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1answer
94 views

Characteristic Class with arbitrary coefficient

I'm looking for some "natural definition" for characteristic class with arbitrary coefficient. For example, the chern class satisfies four conditions and Stiefel-Whitney class satisfies three ...
3
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0answers
43 views

Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
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0answers
18 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 ...
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1answer
51 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 ...
4
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1answer
53 views

Lifting vector bundles along thickenings

Let $X_0 \to X$ be a nilpotent closed immersion of schemes. Is every vector bundle on $X_0$ the pullback of a vector bundle on $X$? The answer is yes when $X$ is affine. In general, there may be ...
1
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1answer
27 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
51 views

Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...
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1answer
93 views

R-linear functionals on manifolds

Surely the following is well known: Let $X$ be a (differentiable) manifold, $R$ the ring of continuous/smooth real functions on $X$, $V$ the $R$-module of all continuous/smooth vector fields on ...
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0answers
39 views

About the geometric interpretation of the reduced external product in K-theory

I'm trying to fill the details of the explanation of the interpretation of the reduced external product. I'll follow Hatcher second books, and I'll write part of his explanation in the gray boxes ...
3
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1answer
83 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
6
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1answer
122 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
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0answers
34 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
3
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1answer
43 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
2
votes
1answer
44 views

“Visual” interpretation of the Bott Periodicity for complex vector bundles

I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ ...
0
votes
1answer
16 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: $[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$ ...
0
votes
3answers
49 views

The difference between a fiber and a section of a vector bundle

If $ E_x := \pi^{-1}(x) $ is the fiber over $x$ where $(E,\pi,M)$ is the vector bundle. And the section is $s: M \to E $ with $\pi \circ s = id_M $. This implies that $\pi^{-1} = s $ on $M$. So then ...
2
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0answers
31 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
2
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0answers
18 views

Euler class of quotient bundle of real projective space

Let $\gamma$ be the real tautological line bundle over the real projective space $\mathbb{R}P^n$, $V$ be the trivial bundle of rank $n+1$, and $Q$ be the quotient bundle $V/\gamma$. What is the Euler ...
4
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1answer
68 views

Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$ TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-)) $$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
2
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1answer
39 views

Working out the details of example 1.13 Hatcher: $\ E_{fg}\oplus n \approx E_f \oplus E_g$

I'm studying an example provided by Hatcher in his K-theory and Vector Bundle book. I'm referring to example 1.13 pag. 24 The first part is clear, $z^2$ is the Kronecker Product of the two ...
3
votes
1answer
62 views

Are vector bundles just modules over $C^{\infty}(M)$, or are “locality” conditions required?

In this question the asker defines 1-forms on a (real, smooth) manifold $M$ to be $C^{\infty}(M)$-module homomorphism[s] from $Vec(M)$ to $C^{\infty}(M).\:\:\:\:(*)$ I'm wondering if this is ...
0
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0answers
28 views

Just a definition of an algebraic bundle, bundles on $\mathbb{P}_n$

I have just realized the notion of an algebraic vector bundle. I have some questions. In particular, I'd like to understand wheather I understand it correctly. Let $$\pi:E\longrightarrow X$$ be a ...
4
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0answers
56 views

Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
4
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0answers
50 views

Holonomy group as a quotient group of $\text{GL}(k,\mathbb{R})$

Currently, I'm reading the script of a Global Analysis lecture at my university. There, we look at a real vector-bundle $(E,\pi,M)$ of rank $k$ with a given connection $\nabla$ and define the ...
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2answers
60 views

An extension of line bundles splits locally

Consider an extension $0\rightarrow L \overset{\alpha}{\rightarrow} E \overset{\beta}{\rightarrow} L' \rightarrow 0$ of bundles and bundle homomorphisms, where $L$ and $L'$ are line bundles. (Let's ...
1
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1answer
47 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
2
votes
1answer
62 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
4
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1answer
41 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 ...
2
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0answers
63 views

Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$ 0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0 $$ Here $VE$ is ...
2
votes
2answers
77 views

Is the following situation about first Chern numbers possible?

Let's consider complex vector bundles on a torus $T^2$ constructed in the following way: Suppose we have a map $f:T^2\to U(n)$, where $U(n)$ is the space of $n\times n$ unitary matrices. This ...
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0answers
24 views

Structures on vector bundles

I am reading the book K theory by Atiyah. In page number 32, he defines some additional structure on a vector bundle $V$. I have understood the definitions there. But there is a statement that says ...
2
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0answers
44 views

Prove the existence (or well-definedness) of the induced connection in tensor bundle

Given a connection $\nabla$ on a vector bundle $E$ over a smooth manifold $M$, we know there is a unique extension of $\nabla$ to all tensor bundles of $E$ that satisfies Leibniz rule and contraction. ...
0
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1answer
35 views

The pullback of a nontrivial line bundle is nontrivial?

Let $X$ be a complex manifold and $E$ a holomorphic vector bundle over $X$ of rank $2$. Let $\mathbb{P}(E)$ denote the projectivization of $E$, with the natural map $p: \mathbb{P}(E) \rightarrow X$. ...
2
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0answers
65 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
2
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0answers
60 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
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0answers
24 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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1answer
41 views

Equivalence between vector field and generator of a group of translations

I've been reading Olver's Applications of Lie groups to differential equations and did not understand the excerp where the autor explains that "[...] every nonvanishing vector field is locally ...
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0answers
20 views

Direct sums, tensor products etc. of $G$-vector bundles are again $G$-spaces

Given two $G$-vector bundles $E$ and $F$ over a $G$-space $X$ ($G$ some finite group), I am interested in the vector bundles $E \oplus F$, $E \otimes F$, $\operatorname{Hom}(E,F)$ etc. I am familiar ...
2
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2answers
90 views

Definition of Normal Bundle

I'm reading Differential Forms in Algebraic Topology by Bott and Tu. I reached the point where the book defines the normal bundle of a submanifold and uses the tubular neighborhood theorem. I can't ...
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0answers
33 views

Homotopic maps induce isomorphic pullbacks (algebraic setting)

I know that in continuous or even smooth category, if $f_i: X \to Y$ is homotopy and $B \to Y$ is vector bundle, than $f_0^* (B) \cong f_1^* (B)$. But I wonder whether there is the same in algebraic ...
0
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1answer
39 views

Affine Bundles vs Affine Spaces

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. ...
2
votes
1answer
43 views

Total space of vector bundle deformation retracts onto 0-section of base space

I'm trying to prove the following: Total space of vector bundle deformation retracts onto 0-section of base space. Books seem to take this fact for granted. I checked Bott Tu and Hatcher. Online ...
0
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0answers
41 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
2
votes
1answer
76 views

Connection vs Curvature

Why is twice a connection usually referred to the curvature: $\overline{\nabla}\circ\nabla=F^\nabla$ Is there an axiomatic definition of curvature, e.g. it is module-linear operator etc?
0
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1answer
43 views

Cocycles vector bundles and metrics

It is well known, and not difficult to prove that a vector bundle $E$ over a (smooth) manifold $M$ together with a metric gives rise to orthonormal frames (by Gram-Schmidt). An consequnece is that the ...