On the Gray index conjecture for phantom maps

We study the Gray index, a numerical invariant for phantom maps. It has been conjectured that the only phantom map between finite-type spaces with infinite Gray index is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid lithe target spaces of the phantom maps are restricted to being simply connected finite complexes. As a result of the counter example, we can show that SNT(infinity)(X) can be non-trivial for some space X of finite type. (C) 2010 Elsevier B.V. All rights reserved.