In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map
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.
[citation needed] If E is an even-graded theory meaning
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
denotes a line passing through x in the underlying vector space
This is the map classifying the tensor product of the universal line bundle over
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).
This topology-related article is a stub.