Statements (49)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
real vector bundles
|
| gptkbp:coefficientRing |
gptkb:Z/2Z
|
| gptkbp:component |
w_1
w_2 w_0 w_n |
| gptkbp:definedIn |
cohomology with Z/2Z coefficients
|
| gptkbp:field |
gptkb:topology
|
| https://www.w3.org/2000/01/rdf-schema#label |
Stiefel-Whitney classes
|
| gptkbp:introducedIn |
1935
|
| gptkbp:namedAfter |
gptkb:Hassler_Whitney
gptkb:Eduard_Stiefel |
| gptkbp:notation |
w_i(E)
|
| gptkbp:property |
functorial
w_1(E) is the obstruction to orientability w_i(E) = 0 for i > rank(E) multiplicative under Whitney sum natural with respect to bundle maps stable under Whitney sum top class detects nontriviality of bundle total Stiefel-Whitney class is sum of all w_i used in classification of manifolds used to distinguish non-isomorphic bundles w_0 is always 1 w_0(E) = 1 for any bundle E w_1 is the first Stiefel-Whitney class w_1(E) = 0 iff E is orientable w_2 is the second Stiefel-Whitney class w_2(E) = 0 iff E admits a spin structure w_2(E) is the obstruction to spin structure w_i of trivial bundle is 0 for i > 0 w_i(E ⊕ F) = sum_{j+k=i} w_j(E)w_k(F) w_i(E) in H^i(X; Z/2Z) w_n is the top Stiefel-Whitney class |
| gptkbp:relatedTo |
gptkb:Pontryagin_classes
gptkb:Chern_classes Euler class |
| gptkbp:type |
characteristic class
|
| gptkbp:usedFor |
differentiable manifolds
obstruction theory classifying vector bundles |
| gptkbp:usedIn |
bundle theory
differential topology homotopy theory cobordism theory |
| gptkbp:bfsParent |
gptkb:Chern_classes
gptkb:Differential_Topology |
| gptkbp:bfsLayer |
6
|