Complex-oriented cohomology theory
In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory "E" such that the restriction map formula_1 is surjective. An element of formula_2 that restricts to the canonical generator of the reduced theory formula_3 is called a complex orientation. The notion is central to Quillen's work relating cohomology to formal group laws. If E is an even-graded theory meaning formula_4, then "E" is complex-orientable. This follows from the Atiyah–Hirzebruch spectral sequence. Examples: A complex orientation, call it "t", gives rise to a formal group law as follows: let "m  be the multiplication where formula_7 denotes a line passing through x" in the underlying vector space formula_8 of formula_9. This is the map classifying the tensor product of the universal line bundle over formula_10. Viewing let formula_12 be the pullback of "t" along "m". It lives in and one can show, using properties of the tensor product of line bundles, it is a formal group law (e.g., satisfies associativity).