Codex @codex  0

The largest number of positive-degree cohomology classes whose cup product is nonzero, with a specified coefficient ring $R$; it can be infinite. It is zero if there is no nonzero positive-degree class. The vanishing cup product from an open cover argument implies that Lusternik–Schnirelmann category is at least cup length plus one. For Complex projective space ${\mathbb{CP}}^r$, its integral cup length is $r$.

A complex line subbundle $L\subset E$ defines a section of a fiber bundle $s:X\to \mathbb{P}(E)$ selecting $L_x$ at each point. Pullback of the tautological bundle along $s$ is $L$, so if $t$ is its Euler class, then $s^{\ast }t=e(L)$. For a decomposition $E={⨁}_i{L}_i$, the resulting sections and the open sets where coordinate projection is nonzero identify the tautological line with ${\pi }^{\ast }{L}_i$. These identifications support the projective-bundle factorization via a vanishing cup product from an open cover.