particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that...

if the second Stiefel–Whitney class w2(M) ∈ H2(M, Z2) of M vanishes. Furthermore, if w2(M) = 0, then the set of the isomorphism classes of spin structures...

geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Let G...

_{2}} -fundamental class. This Z 2 {\displaystyle \mathbf {Z} _{2}} -fundamental class is used in defining Stiefel–Whitney class. If M is a compact orientable...

image of the Euler class under this map is the top Stiefel-Whitney class wr(E). One can view this Stiefel-Whitney class as "the Euler class, ignoring orientation"...

Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class w 9 {\displaystyle...

construction of the Stiefel–Whitney classes of a real vector bundle, thus co-founding the study of characteristic classes. Stiefel entered the Swiss Federal...

In mathematics, the Stiefel manifold V k ( R n ) {\displaystyle V_{k}(\mathbb {R} ^{n})} is the set of all orthonormal k-frames in R n . {\displaystyle...

manifolds have a well-defined Chern class. (See Arakelov geometry) Pontryagin class Stiefel–Whitney class Euler class Segre class Schubert calculus Quantum Hall...

01016. Loomis–Whitney inequality Whitney extension theorem Stiefel–Whitney class Whitney's conditions A and B Whitney embedding theorem Whitney graph isomorphism...

{\displaystyle M} . The first Stiefel–Whitney class classifies smooth real line bundles; in particular, the collection of (equivalence classes of) real line bundles...

form chosen to compute it. It may also be viewed as the second Stiefel–Whitney class of Q. The invariant may be computed for a specific symbol φ taking...

4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class w 2 ( M ) {\displaystyle w_{2}(M)} vanishes), then the signature...

H_{n}(M;\mathbf {Z} )} . A manifold M is orientable if and only if the first Stiefel–Whitney class w 1 ( M ) ∈ H 1 ( M ; Z / 2 ) {\displaystyle w_{1}(M)\in H^{1}(M;\mathbf...

JSTOR 1971304. MR 0808220. Massey, William S. (1960). "On the Stiefel-Whitney classes of a manifold". American Journal of Mathematics. 82 (1): 92–102...

integer coefficients, preserving the linking form and the second Stiefel–Whitney class. Moreover, any such isomorphism in second homology is induced by...

chromatic homotopy theory chromatic homotopy theory. class 1.  Chern class. 2.  Stiefel–Whitney class. classifying space Loosely speaking, a classifying...

Bott and Tu. In his 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these...

/2\mathbf {Z} [w_{1}],} where w 1 {\displaystyle w_{1}} is the first Stiefel–Whitney class: it is the free Z / 2 Z {\displaystyle \mathbf {Z} /2\mathbf {Z}...

normal bundle of M, as detected by its characteristic classes, notably its Stiefel–Whitney classes. That is, since ⁠ R n {\displaystyle \mathbb {R} ^{n}}...

the second Stiefel–Whitney class w 2 ( M ) ∈ H 2 ( M , Z / 2 Z ) {\displaystyle w_{2}(M)\in H^{2}(M,\mathbb {Z} /2\mathbb {Z} )} to a class K ∈ H 2 ( M...

{\mbox{or}}\quad (1,g_{2},g_{3})} depending on the image of the first Stiefel–Whitney characteristic class w 1 {\displaystyle w_{1}} under a Bockstein homomorphism...

cobordism class [ M ] ∈ N n {\displaystyle [M]\in {\mathfrak {N}}_{n}} of a closed unoriented n-dimensional manifold M is determined by the Stiefel–Whitney characteristic...

techniques outlined above and the relation between Chern classes and Stiefel–Whitney classes; note that f : C P 4 ↪ C P 8 {\displaystyle f\colon \mathbf...

Obstruction theory Characteristic class Chern class Chern–Simons form Pontryagin class Pontryagin number Stiefel–Whitney class Poincaré conjecture Cohomology...

MR 3363439. Zbl 0058.13904. Nash, John (1955). "A path space and the Stiefel–Whitney classes". Proceedings of the National Academy of Sciences of the United...

generator of negative degree. Hopf bundle Stiefel-Whitney class Euler sequence Chern class (Chern classes of tautological bundles is the algebraically...

characteristic classes for vector bundles that take values in cohomology, including Chern classes, Stiefel–Whitney classes, and Pontryagin classes. For each...

X × G Y ) {\displaystyle \omega _{n}(X\times _{G}Y)} , the top Stiefel–Whitney class would need to vanish. The identity map i : X → X {\displaystyle...

the first Chern class c 1 ( M ) ∈ H 2 ( M , Z ) {\displaystyle c_{1}(M)\in H^{2}(M,{\mathbb {Z} })} is the second Stiefel-Whitney class w 2 ( M ) {\displaystyle...