Stiefel-Whitney class

From Academic Kids

Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles <math>E\rightarrow X<math>. They are denoted <math>w_i(E)<math>, taking values in <math>H^i(X,\mathbb Z_2)<math>, the cohomology groups with mod <math>2<math> coefficients. Naturally enough, we say that <math>w_i(E)<math> is the <math>i<math>th Stiefel-Whitney class of <math>E<math>. As an example, over the circle, <math>S^1<math>, there is a line bundle that is topologically non-trivial: that is, the line bundle associated to the Möbius band, usually thought of as having fibres <math>[0,1]<math>. The cohomology group

<math>H^1(S^1,\mathbb Z/2\mathbb Z)<math>

has just one element other than <math>0<math>, this element being the first Steifel-Whitney class, <math>w_1<math>, of that line bundle.

Contents

Axioms

Throughout, <math>H^i(\;\cdot\;;G)<math> denotes singular cohomology with coefficient group <math>G<math>.

  1. For every real vector bundle <math>E\rightarrow X<math>, there exist <math>w_i(E)<math> in <math>H^i(X;\mathbb Z/2\mathbb Z)<math> which are natural, i.e., characteristic classes.
  2. <math>w_0(E)=1<math> in <math>H^0(X;\mathbb Z/2\mathbb Z)<math>.
  3. <math>w_ i(E)=0<math> whenever <math>i>\mathrm{rank}(E)<math>.
  4. <math>w_1(\gamma^1)=x<math> in <math>H^1(\mathbb RP^1;\mathbb

Z/2\mathbb Z)=\mathbb Z/2\mathbb Z<math> (normalization condition). Here, <math>\gamma^n<math> is the canonical line bundle.

  1. <math>w_k(E\oplus F)=\sum_{i+j=k}w_i(E)\cup w_j(F)<math>.
  2. If <math>E^k<math> has <math>s_1,\ldots,s_{\ell}<math> sections which are everywhere linearly independent then <math>w_{k-\ell+1}=\cdots=w_k=0<math>.

Some work is required to show that such classes do indeed exist and are unique.

Properties

The first Stiefel-Whitney class is zero if and only if the bundle is orientable.

The second Stiefel-Whitney class is zero if and only if the bundle admits a spin structure.

See also

References

J. Milnor & J. Stasheff, Characteristic Classes, Princeton, 1974.

Template:Math-stub

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