# Questions tagged [stiefel-whitney]

For questions about Stiefel-Whitney classes which are characteristic classes associated to real vector bundles.

14
questions

**5**

votes

**1**answer

255 views

### Using Stiefel-Whitney class to build new principal bundles

I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me.
Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ ...

**3**

votes

**2**answers

273 views

### Is Cohen immersion conjecture (theorem) known for vector bundles?

R. Cohen proved the immersion conjecture in a 1985 Annals paper:
Cohen, Ralph L., The immersion conjecture for differentiable manifolds, Ann. Math. (2) 122, 237-328 (1985). ZBL0592.57022.
Any smooth ...

**7**

votes

**0**answers

231 views

### Different definitions of Stiefel-Whitney classes

It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove ...

**16**

votes

**2**answers

519 views

### What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?

Let $\operatorname{Gr}(k, n)$ and $\operatorname{Gr}^+(k, n)$ denote the unoriented and oriented grassmannians respectively.
The $\mathbb{Z}_2$ cohomology of the unoriented grassmannian is
$$H^*(\...

**10**

votes

**2**answers

497 views

### Is the top Stiefel-Whitney number of a topological manifold the Euler characteristic mod two?

Recall that the Stiefel-Whitney classes of a smooth manifold are defined to be those of its tangent bundle - this definition doesn't extend to topological manifolds as they don't have a tangent bundle....

**17**

votes

**1**answer

1k views

### What are the possible Stiefel-Whitney numbers of a five-manifold?

On a compact five-manifold, the Stiefel-Whitney number $w_2w_3$ can be nonzero.
An example is the manifold $SU(3)/SO(3)$, and also another example is a $\mathbb{CP}^2$ bundle over a circle where the ...

**3**

votes

**2**answers

420 views

### Milnor's proof of cohomology of BO(n)

In Milnor/Stasheff characteristic classes there is the proof that $H^*(BO(n);\mathbb{Z}_2)$ is the polynomial ring on the first n Stiefel-Whitney classes. I understand the part that the latter ring is ...

**8**

votes

**0**answers

212 views

### Dixmier-Douady class is the third integral Stiefel-Whitney class

Let $M$ be (say smooth) manifold. From the short exact sequence of groups $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}_2 \to 0$ (where the first map is multiplication by $2$) one obtain long exact ...

**5**

votes

**0**answers

139 views

### Stiefel-Whitney Class associated to Quadratic Form

I am trying to understand a bit of Milnor's paper "Algebraic K-Theory and Quadratic Forms". I guess, first things first. Why did Milnor come up with such an invariant?
For any number field (such as ...

**4**

votes

**0**answers

151 views

### Obstruction to the existence of lifting of the classifying map

Let $E$ be an $n$-plane bundle over CW complex $X$. Then $E$ is a pullback of tautological bundle $\gamma_n$ over $BO(n)$ i.e. $E=f^*(\gamma_n)$. This $f$ is called classyfing map. One can show that ...

**5**

votes

**2**answers

261 views

### Two set of axioms for Stiefel-Whitney classes

Let $E \to X$ be a vector bundle. We can associate to $E$ several invariants: among them are the Stiefel-Whitney classes $w_i(E) \in H^i(X;\mathbb{Z}_2)$. These classes may be defined using the axioms:...

**8**

votes

**3**answers

537 views

### Stiefel-Whitney total class with prescribed zeros

First things first, I am aware of the existence of this topic. It's related, but old and my question hasn't been discussed there. So I hope it's not wrong to start a new topic.
I'm currently ...

**7**

votes

**2**answers

714 views

### Second Stiefel-Whitney class is a square

I'm interested in examples of manifolds which are orientable and such that the second Stiefel-Whitney class is a square. (Of course the second Stiefel-Whitney class should be non-zero.)
An easy ...

**7**

votes

**2**answers

497 views

### Vector bundle over an oriented manifold with non-vanishing w_2w_3

I am looking for an example of an oriented rank 5 (or lower) real vector bundle $V$ over an oriented manifold such that the cup product $w_2(V) w_3(V)$ of Stiefel-Whitney classes does not vanish. It ...