ELLIPTIC COHOMOLOGY

A generalized cohomology theory is a sequence of contravariant functors {hi}i∈ℤ from spaces to abelian groups which are linked together in a well-known way. The theories that arise in nature are of two types: K-theories, and cobordism theories. (Classical cohomology can be approached in so many different ways that I shall leave it aside for the moment.)

On a compact space X the isomorphism classes of complex vector bundles form an abelian semigroup Vect(X) under the operation of direct sum, and K0(X) is the abelian group got by formally adjoining inverses to the semigroup Vect(X). Then K0 is a homotopy functor, and the functors K−i, for i > 0, defined — roughly — by composing K0 with the i-fold suspension functor, have the properties of “half” a cohomology theory. That much is true for any representable homotopy functor, but the functors Ki are special because of the Bott periodicity theorem, which gives a canonical equivalence between Ki and Ki−2 for i ≤ 0, and enables us to define Ki for all i ∈ ℤ by periodicity.

There is a completely different reason, however, unrelated to Bott periodicity, why the functor K0 forms part of a cohomology theory, and it applies in a much more general context. For any (discrete) ring A we have a contravariant functor X ↦ ModA(X), where ModA(X) is the semigroup of isomorphism classes of bundles of finitely generated projective A-modules on X. It is a representable homotopy functor, though not a very interesting one, as it sees only the fundamental group of X.