Complex-oriented cohomology theory

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

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

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

$E^{*}(\mathbb{ℂ}{\mathbf{𝐏}}^{\mathrm{\infty }})=\underleftarrow{\lim }{E}^{*}(\mathbb{ℂ}{\mathbf{𝐏}}^n)=\underleftarrow{\lim }R[t]/(t^{n+1})=R[[t]],R={\pi }_{*}E$,

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

$E^{*}(\mathbb{ℂ}{\mathbf{𝐏}}^{\mathrm{\infty }}\times \mathbb{ℂ}{\mathbf{𝐏}}^{\mathrm{\infty }})=\underleftarrow{\lim }{E}^{*}(\mathbb{ℂ}{\mathbf{𝐏}}^n\times \mathbb{ℂ}{\mathbf{𝐏}}^m)=\underleftarrow{\lim }R[x,y]/(x^{n+1},y^{m+1})=R[[x,y]]$

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)