Projective bundle relation in complex cobordism ($x^n-p^{\ast }{c}_1(E)x^{n-1}+\cdots +(-1{)}^n{p}^{\ast }{c}_n(E)=0$)

The relation uses $x=c_1(S)$ for the tautological line on the projectivization of $E$. It follows from the rank of the complementary bundle and Whitney multiplication. The alternating signs are ring subtraction; they do not assert that the First Chern class of a dual line is the additive negative in complex cobordism, where duality uses the formal-group inverse.

// Target: geometry-and-topology.bigb