I'm looking for a clarification of the highlighted comment taken from "A User's Guide to Algebraic Topology" by Dodson & Parker. In order to make the setting clear, I uploaded definition $8.1.6$ and the previous corollary for notations. $\chi_{n}\colon \coprod_{\alpha} e_n^{\alpha} \to X^n$ is the union of the characteristic maps which describe the $n$-skeleton.

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And at the successive page (page $260$) they conclude:

This does not deal completely with the problem of extensions since it is possible to climb the whole skeleton with only trivial obstructions, but yet a map may have no extension to $X$; This is the phenomenon of phantom or ghost map

Then they give a reference for these ghost map (Gray Homotopy Theory and McGibbon's paper) but it seems to me that a ghost map is a map whose induced map in homotopy is trivial in every degree but the map itself is not null-homotopic. So I don't see the connection the authors want to make.

More importantly I don't get why the trivial obstructions in every degree are not sufficient for extending the map from $A$ to $X$, with the usual method of passing to the cone of the domain (i.e. applying inductively Corollary $8.1.5$ on all skeleta) in order to extend the map to the entire $X$ which is the colimit of such construction. The comment I highlighted seems to stress some deficiency in my reasoning, but I can't see where I'm wrong.

Can someone clarify?


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