A map $f:X\to Y$ of CW-complexes is called a phantom if $f$ restricted to the $n$-skeleton of $X$ is contractible for all $n$. The first non-trivial example of such a map, with $X=\Sigma\mathbb{P}^\infty(\mathbb{C})$ and $Y$ an infinite wedge of 4-spheres, was constructed by J. F. Adams and G. Walker in 1964. Subsequently, it was shown that many other examples exist. In particular, in 1966 B. Gray showed that there are continuously many non-homotopy equivalent phantoms $K(\mathbb{Z},2)\to S^3$.

I would like to ask if there is a way to construct a non-trivial phantom map, or at least to prove such maps exist, by hand (say, using the material covered in Hatcher's Algebraic topology).

The motivation is this: a colleague of mine has gone away for a while and has asked me if I could replace him during his topology problem classes. I agreed but the instructions given to me were rather vague ("just show them some cool examples..."). I have enough examples to fill all the sessions (a couple of those found on MO by the way), but still I was wondering if I could construct a phantom by hand, and learn myself how to do this. The students seem to be pretty smart but haven't seen much beyond the basic cohomology and homotopy theory, not yet anyway.