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 $X$.
2,659
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Specific homotopy between complex conjugation and the identity.
Consider the set $\mathcal{C} = C^{\infty}(\mathbb{C}^*, \mathbb{C}^*)$, where $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$.
Both $f(z) = z$ and $g(z) = \bar{z}$ can be seen as elements in $\mathcal{C}$...
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How to show $\det(E)\cong M\times \mathbb{R}$ when $M$ is orientable?
If we have an orientable bundle $E\rightarrow M$, then the transition maps can be adjusted by Gram-Schmidt process to be in $SO(n,\mathbb{R})$. So the determinant bundle $\det E$ is isomorphic to $M\...
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How to understand the coordinate transition map in $E\otimes E'$?
This question seems embarrassingly basic. Let $X$ be some manifold and $E,E'$ two vector bundles over $X$ with rank $n$ and $n'$ respectively. Now assume to possible refinement we have $E,E'$ defined ...
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Example of excess intersection theory?
Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ ...
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Complexification of a complex bundle.
Suppose that $E$ is a complex bundle already. This is to say that is can be viewed as a real bundle with an almost complex structure $j$. Then Taubes assert $E$ sits inside its complexification $E_{\...
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Is the trivial bundle ample on an affine variety?
When I was reading a paper, I came across a statement like "Since ince $M$ is affine, the trivial bundle is ample and ..." I think that line bundle $L$ on a variety $M$ is ample if it the global ...
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Trivialisation of the normal bundle of $S^1$
I'd like to show that the normal bundle of $S^1$ is trivial. That is, I want to find a homeomorphism $\varphi : NS^1 \to S^1 \times \mathbb R$ such that $\varphi \mid_{N_s S^1}$ is linear for every $s ...
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The Harder-Narasimhan filtration with inverse slopes.
Let $C$ be a complex curve. Recall that the slope of a coherent sheaf $\mathcal{E}$ is defined by
$$
\mu(\mathcal{E})=\mathrm{Arg}(-\mathrm{deg}(\mathcal{E})+i\mathrm{rank}(\mathcal{E}))\in(0,\pi].
$$
...
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3
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Understanding the trivialisation of a normal bundle
I've been looking for a definition of "trivialisation of normal bundle".
I think I understand what a vector bundle, fibre bundle and a local trivialisation of either is. I also know what a tangent ...
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Question on the definition of ample vector bundles
Let $X$ be a compact complex manifold. According to Fulton and Lazarsfeld, a vector bundle $E$ on $X$ is called ample if the Serre line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ on the projectivized ...
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How do I get the correct coordinate transition functions for the dual bundle?
Suppose $E$ is a vector bundle over $X$, and $X$ is equipped with an atlas of charts $U_{i}$ that is locally finite. Now assume up to possible refinement we have local trivilization $f_{i}: E_{U_{i}}\...
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Why a section of $\otimes_{k}E^{*}$ defines a $k$-linear, fiber preserving map from $\oplus_{k}E$ to $M\times \mathbb{R}$?
I am not sure if this is a duplicate. Clifford Taubes assert in his book Differential Geometry that we may view sections of vector bundles as homomorphisms from $M\times \mathbb{R}$ to $E$ such that ...
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Trivilisations of Vector Bundles
Let $\pi : E \to X$ be a smooth rank $k$ vector bundle on a manifold $X$ (I don't think my question depends upon the stipulations on the bundle, but I've just chosen smooth in case I'm incorrect). By ...
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Classify the vector bundles of a manifold.
I met a question asking me to classify the $2$-dimensional vector bundles of the sphere $S^2$.
I did not know how to classify the vector bundles in general. The only example I know was the line ...
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Does an orientable subbundle of an orientable vector bundle always have a orientable complement?
If I have an orientable vector bundle $E$ and a subbundle $F$ on a manifold $M$, where both the bundles are orientable, does $F$ have a complement in $E$ which is also orientable? Does it have a ...
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Orthogonal complement of a vector bundle
Let $E \rightarrow X$ be a vector bundle with an inner product. If $F$ is a sub-bundle, we can define an orthogonal complement bundle $F^\perp$ (see http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf ...
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Alternate pullback bundle construction
If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is
$$\phi^* F := \{(m,f) \in M \...
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understanding this differential operator on a tensor product
I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
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2
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The Affine Property of Connections on Vector Bundles
Given any two connections $\nabla_1, \nabla_2: \Omega^0 (V) \to \Omega^1 (V)$ on a vector bundle $V \to M$, their difference $\nabla_1 - \nabla_2$ is a $C^\infty (M)$-linear map $\Omega^0 (V) \to \...
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Products of Vector Bundles
Suppose that $E$ is a vector bundle over a compact, Hausdorff space $X$. Then $E^n$ is a vector bundle over $X^n$. If $D(E)$ is the disk bundle, there is a map on fibers $D(E^n)_x \rightarrow D(E)^...
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If a line bundle and its dual both have a section (on a projective variety) does this imply that the bundle is trivial?
Is there any reason that, on a projective variety X, if a line bundle L has a (non-zero) section and also its dual has a section then this implies that L is the trivial line bundle?
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Is Transversality invariant by losing Dimension?
Setup Let $X$ be a smooth, reduced and irreducible scheme. And let $E$ be a vector bundle of rank $n$ on $X$. Following Eisenbud and Harris we say that the collection $\{\sigma_1,...,\sigma_k\}$ of ...
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Orientability of the total space of a vector bundle over an oriented manifold
Let $M$ be a (smooth) manifold of dimension $n$, and let $\pi : E \to M$ be a (smooth) vector bundle of rank $r$. If I choose a connection on $E$, then I obtain a decomposition of $T E$ as $V E \...
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Possible restriction on first nonzero Stiefel-Whitney classes?
I've been reading Hatcher's book on vector bundles and I'm just getting into the section of Steifel-Whitney numbers. Naturally, I'm interested in which sequences of such numbers are realizable, and ...
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Pontryagin classes of a product manifold
I'm imagining there's a way to relate the pontryagin classes of $T(M\times N)$ to the pontryagin classes of $M$ and those of $N$, but I haven't been able to find a helpful reference. Could someone ...
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Why is the tangent bundle orientable?
Let $M$ be a smooth manifold. How do I show that the tangent bundle $TM$ of $M$ is orientable?
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Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?
Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that $(...
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Sum of normal bundle and tangent bundle.
Would somebody be able to prove that the whitney sum of the normal and tangent bundles of a submanifold of $\mathbb{R}^n$ is trivial? Would apreciate a detailed proof...I'm struggling a little.
Tina
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The space of smooth sections of a vector bundle.
Let $M$ be a compact, finite-dimensional manifold and $\pi : \mathrm{B} \rightarrow M$ a vector bundle over $M$ whose typical fiber is $\mathbb{R}^n$. Denote by $\mathcal{C}^{\infty}(M, \mathrm{B})$ ...
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Is the total space of this vector bundle embeddable into $\mathbb{R}^3$?
Let $M$ be the Moebius vector bundle over $S^1$.
Is it possible to embedd the total space of $M\oplus (S^1\times \mathbb{R^1})$ over $S^1$ into $\mathbb{R}^3$?
I suppose this isn't possible but I ...
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Decomposition of vector bundles over a CW complex
Let $X$ be CW complex having only cells up to dimension $n$.
I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a $n$-...
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Example of non-isomorphic vector bundles with the same element in $K$
Let $X$ be a paracompact and well behaved space. Topological K-Theory $K(X)$ of $X$ is group completing the monoid of isomorphism classes of vector bundles over $X$ with the Whitney sum.
Two vector ...
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Vanishing of Chern classes
I finally got the characterization of Chern classes, but i have another question:
Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots ,s_r$ generic global sections, i can characterize ...
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Characterization of Chern classes and Whitney product formula
Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots , s_r$ generic global sections, i can characterize the i-th Chern class as follows:
$ C_i(E)= \eta_{V_i}$, and $V_i$ is the locus ...
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The Canonical Bundle over a Riemann Surface
I am trying to understand an example of a line bundle over a Riemann surface; as it is very terse and short, I have lots of trouble. It is written in block-quotes below, and I ask questions as I go.
...
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Explicit formula for the curvaure of a connection
Let $E$ be a vector bundle over $M$ and denote by $\mathcal{A}^k(E)$ the space of sections of $\Lambda^k (TM)^* \otimes E$, i.e. the space $k$-forms with values in $E$.
A connection $\nabla:\mathcal{...
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Obstruction cocycle of Stiefel manifold
I was reading about the interpretation of Stiefel Whitney classes as obstructions from Milnor-Stasheff's book and I got stuck at a step. The context is the following. Let $E \to B$ be a vector bundle ...
- 1,698
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Chern Classes of a Trivial Bundle
Could someone explain to me why the chern classes of a trivial bundle are zero?
(I'm studying it from Bott & Tu book) To be more specific I can't understand why, given the vector bundle $E$ on $M$...
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Image of smooth vector bundle morphism
Let $\pi_1:V_1 \rightarrow B_1$ and $\pi_2:V_2 \rightarrow B_2$ be smooth vector bundles and write $\phi_V: V_1 \rightarrow V_2$ as well as $\phi_B:B_1 \rightarrow B_2$ respectively for the total and ...
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Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle
Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the
Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the
disjoint union of all these $k$-dimensional subspaces and let
...
- 4,219
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The pullback $F^\ast :T^*N \rightarrow T^*M$ is a smooth bundle map
How can I show that the pullback $F^*: T^*N \rightarrow T^*M$ associated with $F:M \rightarrow N$ is a smooth bundle map if it is a diffeomorphism?
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Real line bundle smoothly isomorphic to Möbius bundle
I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that ...
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Is there a way to establish a correspondence between the vector bundles over a torus and some kind of homotopy groups, just as we do to spheres
By introducing the "clutching function" one can relate the complex (real) vector bundles on a sphere with homotopy groups of $GL_n(\Bbb C)$ ($GL_n^+(\Bbb R)$ for oriented ones). Can we do a similar ...
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Is a sub-bundle of a vector bundle a vector bundle?
Could anyone please help me with this question?
(1) Let $(E, p, B)$ be a vector bundle where $E$ is the total space, $B$ is the base, and $p$ is the structure map, that is, $p:E\to B$. Now suppose $...
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On Frobenius reciprocity theorem
The classical Frobenius reciprocity theorem asserts the following:
If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res U})_{H}.$$...
- 2,258
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Understanding the Definition of a Differential Form of Degree $k$
Let $M$ be a smooth manifold. A differential form of degree $k$ is a smooth section of the $k$th
exterior power of the cotangent bundle of $M$.
Does it mean that a differential form of degree $k$ ...
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Failure of isomorphisms on stalks to arise from an isomorphism of sheaves
It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an ...
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Understanding the canonical line bundle $H$, and the fact that $(H \otimes H)\oplus 1 \simeq H \oplus H$
I am trying to understand Example 1.13 of Hatcher's book on vector bundles and K-Theory (page 24).
The cannonical line bundle $H \to \mathbb{C}P^1$ satisfies the relation $(H \otimes H)\oplus 1 \...
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1
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Tensor Product of Vector Bundles
In Hatcher's book on Vector Bundles he states (page 14) that
It is routine to verify that the tensor product operation for vector bundles over a
fixed base space is commutative, associative, and ...
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Is there a characterization of injective $C(X)$-modules analogous to Serre-Swan?
The Serre-Swan theorem in topology says that if $X$ is compact Hausdorff and $C(X)$ the ring of continuous functions on $X$, then the category of finitely generated projective $C(X)$-modules is ...
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