Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 22 / 1 / c / ii / Solution

A stable map $u:D\to E$ induces a degree-preserving additive natural transformation of represented cohomology theories by postcomposition with ${\Sigma }^{q}u$ in degree $q$. This transformation commutes with the suspension isomorphisms. The resulting mapis onto, but need not be one-to-one. Its kernel consists of hyperphantom maps of spectra: maps $u$ for which $uv=0$ for every stable mapfor every based space $X$ and every $j\ge 0$. Equivalently, they induce zero in all degrees of represented cohomology theory on all based spaces. ThereforeThis condition tests arbitrary suspension spectra with negative shifts. Merely testing finite spectra defines a phantom map of spectra, which is a weaker condition. In particular the homology conclusion in Question 5 should not be substituted for the cohomology condition here.