complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map $E^2(\backslash mathbb\{C\}\backslash mathbf\{P\}^\backslash infty)\; \backslash to\; E^2(\backslash mathbb\{C\}\backslash mathbf\{P\}^1)$ is surjective. An element of $E^2(\backslash mathbb\{C\}\backslash mathbf\{P\}^\backslash infty)$ that restricts to the canonical generator of the reduced theory $\backslash widetilde\{E\}^2(\backslash mathbb\{C\}\backslash mathbf\{P\}^1)$ 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 $\backslash pi\_3\; E\; =\; \backslash pi\_5\; E\; =\; \backslash cdots=0$, 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 $\backslash operatorname\{H\}^2(\backslash mathbb\{C\}\backslash mathbf\{P\}^\backslash infty;\; R)\; \backslash simeq\; \backslash operatorname\{H\}^2(\backslash mathbb\{C\}\backslash mathbf\{P\}^1;R)$.
• 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

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

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

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

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

:$E^*(\backslash mathbb\{C\}\backslash mathbf\{P\}^\backslash infty\; \backslash times\; \backslash mathbb\{C\}\backslash mathbf\{P\}^\backslash infty)\; =\; \backslash varprojlim\; E^*(\backslash mathbb\{C\}\backslash mathbf\{P\}^n\; \backslash times\; \backslash mathbb\{C\}\backslash mathbf\{P\}^m)\; =\; \backslash varprojlim\; R[x,y]/(x^\{n+1\},y^\{m+1\})\; =\; R[\backslash ![x,\; y]\backslash !]$

and one can show, using properties of the tensor product of line bundles, it is a formal group law (e.g., satisfies associativity).