- See also
- References
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.{{citation needed|date=October 2013}}
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. (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
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 (e.g., satisfies associativity).
See also
- Chromatic homotopy theory
References
- M. Hopkins, [https://web.archive.org/web/20150430203224/http://people.virginia.edu/~mah7cd/Foundations/coctalos.pdf Complex oriented cohomology theory and the language of stacks]
- J. Lurie, Chromatic Homotopy Theory (252x)
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