Complex-oriented cohomology theory
In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map is surjective. An element of that restricts to the canonical generator of the reduced theory 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, 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.
- Complex K-theory, denoted KU, is complex-orientable, as it is even-graded.
- 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
where denotes a line passing through x in the underlying vector space of. This is the map classifying the tensor product of the universal line bundle over. Viewing
let 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.
