# Tagged Questions

A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.

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### Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
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### Intuitive explanation for the Atiyah-Singer index theorem

My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem. I'm trying to learn the ...
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### Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

Suppose for some reason one would be expecting a formula of the kind $$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$ valid in $H^*(Y)$ where $f:X\to Y$ is a ...
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### Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello, I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
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### Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah. Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if ...
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### When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
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### Why is it useful to study vector bundles?

I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just ...
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### Symmetric powers and duals of vector bundles in char p

Suppose that $X$ is a smooth projective variety (eg $P^n$) and $E$ is a vector bundle (eg the tangent bundle). If the characteristic is zero, then taking symmetric powers "commutes" with taking duals: ...
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### Using linear algebra to classify vector bundles over P^1

There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let ...
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### vector bundle trivial over every compact subset, then it is globally trivial

Let $X$ be a non-compact metric space (though if the answer to the question is positive, then it probably also holds for more general spaces like, e.g., paracompact Hausdorff) and $E \to X$ a vector ...
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### When can we cancel vector bundles from tensor products?

Let $E,F,G$ be algebraic vector bundles over $\mathbb P_{\mathbb C}^n$. My general question is: Assume $E\otimes G \cong F\otimes G$, under what conditions can one conclude that $E\cong F$? Some ...
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### What are some open problems in algebraic geometry?

What are the open big problems in algebraic geometry and vector bundles? More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...
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### Why is it useful to classify the vector bundles of a space?

It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
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### The third axiom in the definition of (infinite-dimensional) vector bundles: why?

Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
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### If the total Chern class of a vector bundle factors, does it have a sub-bundle?

Motivation: $T_{\mathbb P^2}$ isn't an extension of line bundles Here's a trick to show that the tangent bundle $T$ of $\mathbb P^2$ is not an extension of line bundles. If it were, we would have a ...
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### A question on classification of almost complex structures on $4$-manifolds

I have a (basic?) question in topology. Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by ...
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### Is there an alternative characterisation of vector bundles with vanishing characteristic classes?

This question came up yesterday during our index theory seminar. Let $M$ be a 1-connected smooth manifold and let $E \to M$ be a finite-rank complex vector bundle over $M$. If all the Chern classes ...
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### Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$

I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...
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### Vector bundles vs principal $G$-bundles

It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres $$F=\pi^{-1}(x), \ \ \ x\in B$$ over any $x\in B$, are ...
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### The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get ...
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### Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose ...
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### What is a square root of a line bundle?

If ${L}$ is a line bundle over a complex manifold, what does the square root line bundle $L^{\frac{1}{2}}$ mean? After some google, I got to know that there are certain conditions for the existence of ...
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### Algebraic analogue of the Moebius bundle over the circle

Let $R$ be the ring $R[X,Y]/(X^2+Y^2−1)$. The space of $\mathbb{R}$-rational points of the affine scheme associated to $R$ is the topological circle $S^1$. An algebraic vector bundle over $R$ is an ...
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### Why is the integral of the second chern class an integer?

I'm currently reading the paper "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase" by Barry Simon. Imagine a vector bundle with a connection $\nabla$. For simplicity, we assume that this is ...
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### Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
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