In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map E 2 ( C P ∞ ) → E 2 ( C P 1 ) is surjective. An element of E 2 ( C P ∞ ) that restricts to the canonical generator of the reduced theory E ~ 2 ( C P 1 ) 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 π 3 E = π 5 E = ⋯ , then E is complex-orientable. This follows from the Atiyah–Hirzebruch spectral sequence.
Examples:
An ordinary cohomology with any coefficient ring R is complex orientable, as H 2 ( C P ∞ ; R ) ≃ H 2 ( C P 1 ; R ) .Complex K-theory, denoted KU, is complex-orientable, as it is even-graded. (Bott periodicity theorem)Complex cobordism, whose spectrum is denoted by MU, is complex-orientable.A complex orientation, call it t, gives rise to a formal group law as follows: let m be the multiplication
C P ∞ × C P ∞ → C P ∞ , ( [ x ] , [ y ] ) ↦ [ x y ] where [ x ] denotes a line passing through x in the underlying vector space C [ t ] of C P ∞ . This is the map classifying the tensor product of the universal line bundle over C P ∞ . Viewing
E ∗ ( C P ∞ ) = lim ← E ∗ ( C P n ) = lim ← R [ t ] / ( t n + 1 ) = R [ [ t ] ] , R = π ∗ E ,
let f = m ∗ ( t ) be the pullback of t along m. It lives in
E ∗ ( C P ∞ × C P ∞ ) = lim ← E ∗ ( C P n × C P m ) = lim ← R [ x , y ] / ( x n + 1 , y m + 1 ) = R [ [ x , y ] ] and one can show, using properties of the tensor product of line bundles, it is a formal group law (e.g., satisfies associativity).
