Questions tagged [stiefel-whitney]
For questions about Stiefel-Whitney classes which are characteristic classes associated to real vector bundles.
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Examples of manifolds with first nontrivial SW-class in degree 16 or bigger
As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi ...
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Pullback of $w_1$ for 3-manifolds
Given closed $3$-manifolds $M$ and $N$
and an element $\alpha\in H^1(M;\mathbb{Z}_2)$,
when does there exist a map $f:M\to N$
such that $\alpha=f^*(w_1(N))$?
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Stiefel Whitney number of a fiber bundle
I was going through this paper, and the author rights the following
The Stiefel-Whitney class of $E$ is given by $$w(E)=(1+\alpha)^{2m+1}\left\{(1+c)^{2n+1}+u_1(1+c)^{2n}+\dots+u_{2n}(1+c)+u_{2n+1}\...
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What characteristic classes are there?
Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(...
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Which Stiefel-Whitney numbers can be extended to manifolds with boundaries?
The Stiefel-Whitney numbers are classical topological manifold invariants obtained by integrating some local quantity (a cup product of Stiefel-Whitney classes) over the manifold. Which Stiefel-...
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Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$
For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e.,
$$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
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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 $w_{2^k}(E)$ (...
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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 ...
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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 ...
user131113
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Analogy between Stiefel-Whitney and Chern classes
There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field $\mathbb R$ with $\mathbb C$, coefficient ring $\mathbb Z/2$ with $\mathbb Z$ and scales the grading ...
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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 ...
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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,\...
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Which elements of $H^2(M,\mathbb{Z}/2)$ are the $w_2(E)$ for a real bundle $E$?
Any element of $H^1(M,\mathbb{Z}/2)$ is the $w_1(E)$ of a real line bundle $E$ over $M$.
I wonder how to characterize (probably using the Steenrod squares) which elements of $H^2(M,\mathbb{Z}/2)$ are ...
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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 $\Sigma$ ...
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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 ...
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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 ...
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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^*(\...
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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....
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
- 8,786
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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:...
- 8,786
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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 ...
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Who discovered this definition of Stiefel-Whitney classes?
I would define Stiefel-Whitney classes as the pullbacks of generators of $H^*(BO, \mathbb{Z}/2)$ under a classifying map, and I gather this is a pretty common definition.
However, the book "...
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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 ...
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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 ...
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