Complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map $E^{2}({\mathbb {C}}{\mathbf {P}}^{\infty })\to E^{2}({\mathbb {C}}{\mathbf {P}}^{1})$ is surjective. An element of $E^{2}({\mathbb {C}}{\mathbf {P}}^{\infty })$ that restricts to the canonical generator of the reduced theory $\widetilde {E}^{2}({\mathbb {C}}{\mathbf {P}}^{1})$ 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 $\pi _{3}E=\pi _{5}E=\cdots$ , 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 $\operatorname {H}^{2}({\mathbb {C}}{\mathbf {P}}^{\infty };R)\simeq \operatorname {H}^{2}({\mathbb {C}}{\mathbf {P}}^{1};R)$ .
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

{\mathbb {C}}{\mathbf {P}}^{\infty }\times {\mathbb {C}}{\mathbf {P}}^{\infty }\to {\mathbb {C}}{\mathbf {P}}^{\infty },([x],[y])\mapsto [xy]

where $[x]$ denotes a line passing through x in the underlying vector space ${\mathbb {C}}[t]$ of ${\mathbb {C}}{\mathbf {P}}^{\infty }$ . This is the map classifying the tensor product of the universal line bundle over $\mathbb {C} \mathbf {P} ^{\infty }$ . Viewing

E^{*}(\mathbb {C} \mathbf {P} ^{\infty })=\varprojlim E^{*}(\mathbb {C} \mathbf {P} ^{n})=\varprojlim R[t]/(t^{n+1})=R[\![t]\!],\quad R=\pi _{*}E

let $f=m^{*}(t)$ be the pullback of t along m. It lives in

E^{*}({\mathbb {C}}{\mathbf {P}}^{\infty }\times {\mathbb {C}}{\mathbf {P}}^{\infty })=\varprojlim E^{*}({\mathbb {C}}{\mathbf {P}}^{n}\times {\mathbb {C}}{\mathbf {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).

References

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

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Complex-oriented cohomology theory

See also

References