Complex-oriented cohomology theory explained

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map

E ^{2(CP} ^{infty)} \to E ^{2(CP} ^{1)} is surjective. An element of E ^{2(CP} ^{infty)} that restricts to the canonical generator of the reduced theory \widetilde{E} ^{2(CP} ^{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

\pi _{3} E = \pi _{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

\operatorname{H} ^{2(CP} ^{infty;} R) \simeq \operatorname{H} ^{2(CP} ^{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 ^{infty} x C P ^{infty} \to C P ^{infty,} ([x], [y]) \mapsto [xy] where [x] denotes a line passing through x in the underlying vector space C [t] of C P ^{infty} . This is the map classifying the tensor product of the universal line bundle over C P ^{infty} . ViewingE ^{*(CP} ^{infty)} = \varprojlim E ^{*(CP} ^{n)} = \varprojlim R[t]/(t ^{n+1} ) = R[[t]], R = \pi _{*} E ,let f = m ^{*(t)} be the pullback of t along m. It lives inE ^{*(CP} ^{infty} x C P ^{infty)} = \varprojlim E ^{*(CP} ^{n} x C P ^{m)} = \varprojlim 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).

See also

• Chromatic homotopy theory

References

• M. Hopkins, Complex oriented cohomology theory and the language of stacks
• J. Lurie, Chromatic Homotopy Theory (252x)