A homology theory consists
of a sequence {hn} of covariant functors from a suitable category of C∗-algebras to
abelian groups which satisfies homotopy and exactness axioms. We show that such
theories have Mayer-Vietoris sequences and (if additive) commute with inductive
limits. There are analogous definitions and theorems in cohomology with one
important difference: an additive cohomology theory associates a Milnor lim1
sequence to an inductive limit of C∗-aIgebras. As prerequisite to these results we
develop the necessary homotopy theory, including cofibrations and cofibre
theories.
