Complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map ${\displaystyle E^{2}(\mathbb {C} \mathbf {P} ^{\infty })\to E^{2}(\mathbb {C} \mathbf {P} ^{1})}$ is surjective. An element of ${\displaystyle E^{2}(\mathbb {C} \mathbf {P} ^{\infty })}$ that restricts to the canonical generator of the reduced theory ${\displaystyle {\widetilde {E}}^{2}(\mathbb {C} \mathbf {P} ^{1})}$ is called a complex orientation. The notion is central to Quillen's work relating cohomology to formal group laws.

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