Questions tagged [stiefel-whitney]

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

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5
votes
1answer
94 views

Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$

For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e., $$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
9
votes
0answers
158 views

Realizing Stiefel-Whitney classes via vector bundles

Let $X$ be a CW complex. If $E$ is a vector bundle over $X$, then it's well-known that the Stiefel-Whitney classes $w_j(E) \in H^j(X,\mathbb F_2)$ of $E$ are determined from the classes $w_{2^k}(E)$ (...
7
votes
1answer
186 views

Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?

This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case. Let $M$ be a ...
6
votes
1answer
301 views

Action of Steenrod algebra on Chern classes

This is question about result from Brown and Peterson $H^*(MO)$ as an algebra over the Steenrod algebra. Unfortunately, the paper is not available on the Internet, so I can't find the proof. One of ...
2
votes
0answers
128 views

If $n$ is not a power of 2 then the dual Stiefel-Whitney class $\bar{w}_{n-1} = 0$

Stiefel-Whitney classes are invertible and for $w$, the Stiefel-Whitney class of the tangent bundle of $M$, we have its inverse $\bar{w}$. I want to prove that if $n$ is not a power of 2 then the dual ...
5
votes
0answers
118 views

Clarify formula for Steifel-Whitney (Poincaré dual) homology classes in a barycentric subdivision?

Let $X$ be a triangulated manifold of dimension $n$. Let $[W_{n-p}] \in H_{n-p}(X,\mathbb{Z}_2)$, be the homology class that's Poincaré dual to the $p$-th Stiefel-Whitney class $[w_p] \in H^p(X,\...
5
votes
1answer
321 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
2answers
321 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
0answers
260 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
2answers
648 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^*(\...
12
votes
3answers
626 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....
19
votes
1answer
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
2answers
548 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
0answers
226 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
0answers
154 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
0answers
156 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
2answers
291 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
3answers
554 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
2answers
813 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
2answers
587 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 ...