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Complex-oriented cohomology theory

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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.[citation needed]

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).

See also

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References

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