Stiefel-Whitney class: Difference between revisions

Revision as of 22:03, 24 December 2007

Definition

The Stiefel-Whitney class is a characteristic class 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.

@@ Line 6: / Line 6: @@
 # <math>w_i(f^*(E)) = f^*(w_i(E))</math> where <math>f^*</math> denotes the pullback (this is the condition for being a [[natural transformation]], and is  part of the definition of a [[characteristic class]])
-# <math>w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)</math>
+# <math>w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)</math> (this is equivalent to the [[Whitney sum formula]])
 # <math>w_i(E) = 0</math> if <math>i</math> is greater than the dimension of <math>E</math>
 * For the canonical line bundle <math>E \to \R P^\infty</math>, <math>w_1(E)</math> is a generator of <math>H^1(\R P^\infty; \mathbb{Z}_2)</math>
 <math>w</math> is termed the '''total Stiefel Whitney-class''' and <math>w_i</math> is termed the <math>i^{th}</math> Stiefel-Whitney class.