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- In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map is surjective. An element of that restricts to the canonical generator of the reduced theory 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 , 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 .
* 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 where denotes a line passing through x in the underlying vector space of . This is the map classifying the tensor product of the universal line bundle over . Viewing , let be the pullback of t along m. It lives in and one can show, using properties of the tensor product of line bundles, it is a formal group law (e.g., satisfies associativity). (en)
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- 2722 (xsd:nonNegativeInteger)
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- In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map is surjective. An element of that restricts to the canonical generator of the reduced theory 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 , then E is complex-orientable. This follows from the Atiyah–Hirzebruch spectral sequence. Examples: A complex orientation, call it t, gives rise to a formal group law as follows: let m be the multiplication , (en)
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- Complex-oriented cohomology theory (en)
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