Princípio de divisão

Em matemática, o princípio de divisão é uma técnica usada para reduzir questões sobre fibrados vectoriais para o caso de fibrados de linhas.

Em sua forma mais comum, o princípio pode ser enunciado do seguinte modo: ^[1]

Teorema: Seja $\xi \colon E\rightarrow X$ um fibrado vetorial de dimensão $n$ sobre um espaço paracompacto $X$ . Então existe uma variedade $Y=Fl(E)$ e uma aplicação $p\colon Y\rightarrow X$ tal que

o homomorfismo induzido na cohomologia $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ é injetivo e
o pullback $p^{*}\xi \colon p^{*}E\rightarrow Y$ se divide como a soma direta de fibrados de linha: $p^{*}(E)=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{n}.$

As classes de Chern $c_{1}(L_{1}),...,c_{n}(L_{n})$ são ditas as raízes de Chern de $E$ . O ponto é que, como $p^{*}$ é injetiva, toda fórmula envolvendo classes de Chern em $Y$ vale também em $X$ . Para provarmos fórmulas do tipo, portanto, podemos considerar somente somas diretas de fibrados de linha.

O princípio da divisão possui várias variações. A seguinte, em particular, trata de fibrados vetoriais reais e suas complexificações:^[2]

Teorema: Seja $\xi \colon E\rightarrow X$ um fibrado vetorial real de dimensão $2n$ sobre um espaço paracompacto $X$ . Então existe um espaço $Y$ e uma aplicação $p\colon Y\rightarrow X$ tal que

o homomorfismo induzido na cohomologia $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ é injetivo e
o pullback $p^{*}\xi \colon p^{*}E\rightarrow Y$ se divide como a soma de fibrados de linha: $p^{*}(E\otimes \mathbb {C} )=L_{1}\oplus {\overline {L_{1}}}\oplus \cdots \oplus L_{n}\oplus {\overline {L_{n}}}.$

Referências

↑ Raoul Bott e Loring W. Tu, Differential Forms in Algebraic Toplogy, seção 21.
↑ H. Blane Lawson e Marie-Louise Michelsohn, Spin Geometry, Proposição 11.2.

Hatcher, Allen (2003), Vector Bundles & K-Theory (2.0 ed.) seção 3.1

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