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Template:Orphan In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map $E^{2} (ℂ 𝐏^{\infty}) \to E^{2} (ℂ 𝐏^{1})$ is surjective. An element of $E^{2} (ℂ 𝐏^{\infty})$ that restricts to the canonical generator of the reduced theory ${\tilde{E}}^{2} (ℂ 𝐏^{1})$ is called a complex orientation. The notion is central to Quillen's work relating cohomology to formal group laws.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

If $π_{3} E = π_{5} E = \dots$ , then E is complex-orientable.

Examples:

An ordinary cohomology with any coefficient ring R is complex orientable, as $H^{2} (ℂ 𝐏^{\infty}; R) ≃ H^{2} (ℂ 𝐏^{1}; R)$ .
A complex K-theory, denoted by K, is complex-orientable, as $π_{3} K = π_{5} K = \dots = 0.$ (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} \times ℂ 𝐏^{\infty} \to ℂ 𝐏^{\infty}, ([x], [y]) \mapsto [x y]

where $[x]$ denotes a line passing through x in the underlying vector space $ℂ [t]$ of $ℂ 𝐏^{\infty}$ . Viewing

E^{*} (ℂ 𝐏^{\infty}) = \lim_{\leftarrow} E^{*} (ℂ 𝐏^{n}) = \lim_{\leftarrow} R [t] / (t^{n + 1}) = R [[t]], R = π_{*} E = \oplus π_{2 n} E

,

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

E^{*} (ℂ 𝐏^{\infty} \times ℂ 𝐏^{\infty}) = \lim_{\leftarrow} E^{*} (ℂ 𝐏^{n} \times ℂ 𝐏^{m}) = \lim_{\leftarrow} R [x, y] / (x^{n + 1}, y^{m + 1}) = R [[x, y]]

and one can show it is a formal group law (e.g., satisfies associativity).

References