Complex-Oriented Cohomology Theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map [math]\displaystyle{ E^2(\mathbb{C}\mathbf{P}^\infty) \to E^2(\mathbb{C}\mathbf{P}^1) }[/math] is surjective. An element of [math]\displaystyle{ E^2(\mathbb{C}\mathbf{P}^\infty) }[/math] that restricts to the canonical generator of the reduced theory [math]\displaystyle{ \widetilde{E}^2(\mathbb{C}\mathbf{P}^1) }[/math] 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 [math]\displaystyle{ \pi_3 E = \pi_5 E = \cdots }[/math], 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 [math]\displaystyle{ \operatorname{H}^2(\mathbb{C}\mathbf{P}^\infty; R) \simeq \operatorname{H}^2(\mathbb{C}\mathbf{P}^1;R) }[/math].
• 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

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

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

[math]\displaystyle{ 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 }[/math],

let [math]\displaystyle{ f = m^*(t) }[/math] be the pullback of t along m. It lives in

[math]\displaystyle{ 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]\!] }[/math]

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)

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