# 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.

3k views

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

Are the Milnor's seven dimensional exotic spheres parallelizable?
6k views

### 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 ...
3k views

### 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 ...
3k views

### 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 ...
659 views

### 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 ...
2k views

### 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 ...
6k views

### 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 ...
3k views

2k views

### First Chern class of a flat line bundle

A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one? Let $X$ be a nice space (...
214 views

### “High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
1k views

### 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 ...
5k views

### Why is cotangent more canonical than tangent?

You don't need a metric to define the differential of a function, and the cotangent bundle carries a canonical one-form. But you do need a metric to define the gradient, and the tangent bundle does ...
1k views

### A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
1k views

### How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best possible:...
2k views

### Is the category of vector bundles over a topological space abelian?

Consider the trivial bundle $V=\mathbb{R}\times\mathbb{R}$ and the map $f:V\rightarrow V$ given by $(t,x)\mapsto(t,tx)$. This has fibrewise kernels and cokernels, but the ranks jump at 0, so the ...
593 views

### Existence of non-split vector bundles on smooth projective varieties

Question. Is it known/easy to see that every smooth projective variety $X$ (over an algebraically closed field), except for the point and $\mathbb{P}^1$, has a vector bundle which is not a direct sum ...
510 views

### Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
3k views

### Extending vector bundles on a given open subscheme

Many people seem to know the following. Personally, I don't quite understand it though and maybe I'm wrong. It's the fact that "a vector bundle on an open subscheme extends in only one way to a vector ...
2k views

### Representations of surface groups via holomorphic connections

EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it! Background Let $E \to X$ be a ...
1k views

### Non trivial vector bundle over non-paracompact contractible space

The proof that the set of classes of vector bundles is homotopy invariant relies on the paracompactness and the Hausdorff property of the base space. Are there any known examples of: Non trivial ...
3k views

### Evidences on Hartshorne's conjecture? References?

Hartshorne's famous conjecture on vector bundles say that any rank $2$ vector bundle over a projective space $\mathbb{P}^n$ with $n\geq 7$ splits into the direct sum of two line bundles. So my ...
593 views

### Soft and hard part of geometry [duplicate]

While listening to some lecture of Alain Connes about noncommutative geometry, he spoke about various generalizations of the classical concepts from geometry and divided it into "soft" and "hard" part....
202 views

### Smooth manifolds as idempotent splitting completion

The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps. ...
1k views

### Does every vector bundle allow a finite trivialization cover?

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$. (a) Is ...
1k views

### what is a spinor structure?

There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$, a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$ ...
2k views

### What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
840 views

845 views

### Splitting principle for holomorphic vector bundles

Let $E \to X$ be a vector bundle over a decent space $X$. Then there is a space $Z$ together with a map $p: Z \to X$ which induces a split injection on cohomology and such that $p^* E$ splits as a ...
1k views

### 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 ...
908 views

### 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 ...
1k views

### K-Theory and the Stack of Vector Bundles

I have some understanding that vector bundles provide a basic, familiar example of what I should call a stack. Namely, consider the functor $Vect$ that assigns to a space $X$ the set of isomorphism ...
255 views

### Do vanishing characteristic classes of the tangent bundle imply a manifold is stably frameable?

Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, ...
870 views

### Flat SU(2) bundles over hyperbolic 3-manifolds

Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold? The literature on such bundles over 3-manifolds is huge and my naive searches don'...
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 \$...