Stiefel-Whitney class: Difference between revisions

Revision as of 22:04, 24 December 2007

Definition

The Stiefel-Whitney class is a characteristic class (or collection of characteristic classes in different dimensions) for the topological group $G L (n, R)$ with coefficients mod 2. It can be defined axiomatically as follows.

To each real vector bundle $p : E \to B$ , an element $w \in H^{*} (B; Z_{2})$ such that if $w_{i}$ denotes the component of $w$ in $H^{i} (B; Z_{2})$ , we have:

$w_{i} (f^{*} (E)) = f^{*} (w_{i} (E))$ where $f^{*}$ denotes the pullback (this is the condition for being a natural transformation, and is part of the definition of a characteristic class)
$w (E_{1} \oplus E_{2}) = w (E_{1}) ⌣ w (E_{2})$ (this is equivalent to the Whitney sum formula)
$w_{i} (E) = 0$ if $i$ is greater than the dimension of $E$

For the canonical line bundle $E \to R P^{\infty}$ , $w_{1} (E)$ is a generator of $H^{1} (R P^{\infty}; Z_{2})$

$w$ is termed the total Stiefel Whitney-class and $w_{i}$ is termed the $i^{t h}$ Stiefel-Whitney class.

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 ==Definition==
-The Stiefel-Whitney class is a [[characteristic class]] for the topological group <math>GL(n,\R)</math> with coefficients mod 2. It can be defined axiomatically as follows.
+The Stiefel-Whitney class is a [[characteristic class]] (or collection of characteristic classes in different dimensions) for the topological group <math>GL(n,\R)</math> with coefficients mod 2. It can be defined axiomatically as follows.
 To each [[real vector bundle]] <math>p:E \to B</math>, an element <math>w \in H^*(B;\mathbb{Z}_2)</math> such that if <math>w_i</math> denotes the component of <math>w</math> in <math>H^i(B;\mathbb{Z}_2)</math>, we have: