TITLE: Conjuring phantoms by hand?
QUESTION [25 upvotes]: 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.

REPLY [12 votes]: For any CW complex $X$ there is a map $\theta_X:X\to \bigvee_{n=1}^\infty \Sigma X_n$, where $X_n$ denotes the $n$-skeleton of $X$. This map can be realized as the cofiber of the folding map $\bigvee_{n=1}^\infty X_n \to X$. By construction, the map $\theta_X$ is phantom, called the Universal Phantom Map out of $X$. 
Gray showed in his thesis that every phantom map $X\to Y$ factors through $\theta_X$ up to homotopy, so essential phantom maps out of $X$ exist if and only if $\theta_X$ is nontrivial. Since we know, for example, that $\mathbb{C}P^\infty$ is the domain of essential phantom maps, it follows that $\theta_{\mathbb{C}P^\infty}:\mathbb{C}P^\infty\to \bigvee \Sigma \mathbb{C}P^n$ is an essential phantom map that is the homotopy cofiber of the map $\bigvee \mathbb{C}P^n \to \mathbb{C}P^\infty$. 
This example is included in McGibbon's article in the Handbook, and is addressed in some detail in Modern Classical Homotopy Theory by Jeff Strom.