Phantom maps and chromatic phantom maps

		 J. Daniel Christensen and Mark Hovey

Keywords: Phantom map, chromatic phantom map, n-phantom map,
          cohomotopy, stable homotopy, spectrum, n-finite type.

Abstract:

  In the first part, we determine conditions on spectra X and Y under
  which either every map from X to Y is phantom, or no nonzero maps are.
  We also address the question of whether such all or nothing behaviour
  is preserved when X is replaced with V smash X for V finite.  In the
  second part, we introduce chromatic phantom maps.  A map is n-phantom
  if it is null when restricted to finite spectra of type at least n.
  We define divisibility and finite type conditions which are suitable
  for studying n-phantom maps.  We show that the duality functor W_{n-1}
  defined by Mahowald and Rezk is the analog of Brown-Comenetz duality
  for chromatic phantom maps, and give conditions under which the
  natural map Y --> W_{n-1}^2 Y is an isomorphism.