Characteristic number

In the mathematical field of algebraic topology , characteristic numbers are defined by applying combinations of characteristic classes to the fundamental class of a manifold . Pontryagin numbers and Stiefel-Whitney numbers are of particular importance .

Boots Whitney Figures

Let it be a -dimensional differentiable manifold and its tangential bundle . For every partition of (i.e. every decomposition as the sum of positive integers) one has a Stiefel-Whitney number ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle TM}$ ${\displaystyle n}$ ${\displaystyle n=n_{1}+\ldots +n_{k}}$

${\displaystyle \langle w_{n_{1}}(TM)\cap \ldots \cap w_{n_{k}}(TM),\,\left[M\right]\rangle \in \mathbb {Z} /2\mathbb {Z} }$,

where the -th Stiefel-Whitney class of the tangential bundle denotes the cup product , the fundamental class and the Kronecker pairing . ${\displaystyle w_{i}(TM)\in H^{i}(M;\mathbb {Z} /2\mathbb {Z} )}$ ${\displaystyle i}$ ${\displaystyle \cap }$ ${\displaystyle \left[M\right]\in H_{n}(M;\mathbb {Z} /2\mathbb {Z} )}$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

Pontryagin numbers

Let it be an orientable , -dimensional, differentiable manifold and its tangential bundle . For every partition of one has a Pontryagin number ${\displaystyle M}$ ${\displaystyle 4n}$ ${\displaystyle TM}$ ${\displaystyle n}$

${\displaystyle \langle p_{n_{1}}(TM)\cap \ldots \cap p_{n_{k}}(TM),\,\left[M\right]\rangle \in \mathbb {Z} }$,

where the -th Pontryagin class of the tangential bundle denotes the cup product, the fundamental class and the Kronecker pairing. ${\displaystyle p_{i}(TM)\in H^{4i}(M;\mathbb {Z} )}$ ${\displaystyle i}$ ${\displaystyle \cap }$ ${\displaystyle \left[M\right]\in H_{4n}(M;\mathbb {Z} )}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

literature

• John Milnor , James Stasheff : Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974.