# Tagged Questions

**1**

vote

**1**answer

199 views

### Possible homotopy-theoretical approach to Gauss-Bonnet

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 ...

**15**

votes

**1**answer

390 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 ...

**9**

votes

**1**answer

352 views

### Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles?
Can you give me a text or an article?
For example:
Consider a vector bundle $E$ with fiber $V$ and base manifold $M$.
Consider ...

**-3**

votes

**1**answer

256 views

### Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...

**1**

vote

**2**answers

223 views

### A generalized Thom Isomorphism

The ordinary Thom isomorphism says $H^{*+n}(E,E_{0}) \simeq H^{*}(X)$, where $E$ is a vector bundle over $X$ and $E_{0}$ is $E$ minus the zero section. Now assume that $S$ is a non vanishing section ...

**2**

votes

**2**answers

161 views

### Canonical n plane bundle over Lagrangian Grassmanian

Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...

**-1**

votes

**2**answers

161 views

### Restriction of a line bundle to a two-cycle

I am reading a paper on Chiral Differential Operators
http://arxiv.org/pdf/hep-th/0604179v3.pdf
and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...

**2**

votes

**1**answer

160 views

### Stiefel classes and generic sections

I asked this question in math.stackexchange few days ago.
Unfortunately, I haven't seen any simple answer.
One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly ...

**25**

votes

**2**answers

889 views

### 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 ...

**1**

vote

**0**answers

469 views

### Cech Cohomology of Sections of Holomorphic Bundle over Contractible Space

Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$?
I know that such a bundle must ...

**1**

vote

**1**answer

234 views

### non-trivial topological line bundles over cartesian product of manifolds not coming from a pullback

Let $X$ and $Y$ be connected smooth manifolds. Let $L$ be a topological real line
bundle over $X\times Y$. Then we know that the isomorphism class of such a line bundle is determined by its first ...

**5**

votes

**3**answers

426 views

### How does one go from Chern--Weil to cohomology classes on BGL(n,C)?

Let's assume we start with Chern--Weil theory in the following form:
Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l_n(\mathbb C)$ ...

**13**

votes

**2**answers

829 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 ...

**7**

votes

**2**answers

315 views

### References/surveys concerning characteristic classes of flat vector bundles

I'm looking for good surveys about characteristic classes of flat real vector bundles. Letting $G$ be $\text{SL}_n(\mathbb{R})$ with the discrete topology, orientable flat $n$-dimensional real vector ...

**4**

votes

**1**answer

452 views

### etale homotopy and Adams conjecture

I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, ...

**12**

votes

**6**answers

783 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 ...

**9**

votes

**2**answers

524 views

### Configuration spaces and non homeomorphic vector bundles

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.
...

**35**

votes

**4**answers

2k 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 ...

**3**

votes

**2**answers

524 views

### Stiefel-Whitney classes of a projective space bundle

Hi!
Let $\gamma_1$ denote the twisted line bundle over $S^1$ and add a trivial $(2k-1)$-bundle $\mathbb{R}^{2k-1}$. Consider the projective bundle $P(\gamma_1 \oplus \mathbb{R}^{2k-1})$ over $S^1$. ...

**12**

votes

**1**answer

569 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 ...

**8**

votes

**1**answer

453 views

### Weak Vector Bundles

The following notion has arisen in a paper I'm writing.
Definition. A map $p: E\to B$ of spaces
is said to be weak vector bundle if for all compact subspaces $K \subset B$
the restriction of $p$ to
...

**15**

votes

**0**answers

460 views

### Characteristic Classes for $E_8$ Bundles

Given a principal $E_8$ bundle $P\rightarrow X$ one can take the
adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$
and form the associated vector bundle $V=P\times_{\rho}\mathbb
...

**27**

votes

**3**answers

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 ...

**8**

votes

**2**answers

878 views

### Nice example of a topologically trivial bundle with nontrivial connection

So, I've been trying to understand what exactly an anomaly is, and how they arise in physics. Apparently an anomalous theory is some theory whose action is given by a section of some bundle (rather ...

**4**

votes

**0**answers

292 views

### Vector bundles of schemes and their topological realizations

Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$.
Does $R_\mathbb{R}$ send an ...

**12**

votes

**1**answer

907 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 ...

**6**

votes

**2**answers

623 views

### Which torsion classes in integral cohomology are Chern classes of flat bundles?

Chern-Weil theory tells us that the integral Chern classes of a flat bundle over a compact manifold (i.e. a bundle admitting a flat connection) are all torsion. Given a compact manifold $M$ whose ...

**14**

votes

**2**answers

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 ...

**12**

votes

**1**answer

651 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 ...

**7**

votes

**2**answers

341 views

### Swan like theorem and covering spaces

Let $X$ be a finite CW complex. Swan's theorem provide an equivalence
\[
Vec(X)~\xrightarrow{\sim} ~ProjMod(hom_{Top}(X,\mathbb{R}))
\]
between the category of ...

**8**

votes

**4**answers

608 views

### Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold?

EDIT: The original question was answered very quickly (and very nicely!) but the answer leads to a pretty obvious subsequent question, which I will now ask. The original question is maintained for ...

**6**

votes

**5**answers

1k views

### Vanishing of Euler class

Given a real oriented vector bundle E over the base space B of rank n, such that the Euler characteristic class in the n-th cohomology group of B vanishes, is it true that there exists a global ...

**16**

votes

**5**answers

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 ...

**3**

votes

**1**answer

771 views

### Quick ways to calculate cohomology of vector bundle/local system from transition functions?

Suppose I have a vector bundle (or local system, or something else given by transition functions) on a Riemann surface (or generally a (complex) manifold), and I want to compute its cohomology. The ...

**5**

votes

**1**answer

654 views

### Maslov index of a pullback bundle

This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology.
Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex ...

**10**

votes

**2**answers

914 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 ...

**3**

votes

**2**answers

713 views

### How to prove that w_1(E)=w_1(detE) ?

How to prove that the first Stiefel-Whitney class $w_1 (E)$ of a real rank $n$ vector bundle over a manifold M is equal to $w_1(\operatorname{det} E)$, where $\operatorname{det} E$ is the $n$-th wedge ...

**5**

votes

**1**answer

369 views

### Uniqueness of Chern/Stiefel-Whitney Classes

This question is closely related to this previous question.
Chern and Stiefel-Whitney classes can be defined on bundles over arbitrary base spaces. (In Hatcher's Vector Bundles notes, he uses the ...

**4**

votes

**3**answers

2k views

### Grassmannian bundle theorem

Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E_x$ by $Gr(k,E_x)$.
...