14 - What is an elliptic object?

Summary

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 ↦ Mod_A(X), where Mod_A(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.