Cohomological Bousfield classes Original Research Article

In this paper, we begin the study of Bousfield classes for cohomology theories defined on spectra. Our main result is that a map f:X → Y induces an isomorphism on E(n)-cohomology if and only if it induces an isomorphism on E(n)-homology. We also prove this for variants of E(n) such as elliptic cohomology and real K-theory. We also show that there is a nontrivial map from a spectrum Z to the K(n)-local sphere if and only if K(n)*(Z) ≠ 0.