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This question comes from the text Stable Homotopy and Generalised Homology by Adams and so the "stable homotopy category" here refers to his construction via CW-spectra. With this framework in mind, at one point Adams defines the mapping cone of an arrow $f:E \rightarrow F$ by first choosing a representative $f':E' \rightarrow F$ (where $E'$ is a cofinal subspectrum) and then constructing the $n$th term of this spectrum by setting $F_n \cup_{f'_n} CE_n'$. One then shows this is effectively independent of our initial choice of cofinal subspectrum and the argument is complete. One point in this argument is not entirely clear to me, however, and I would like some further clarification:

Why $F_n \cup_{f'_n} CE_n'$ is necessarily a CW-complex?

The first obstacle is that $f'_n$ need not even be cellular, but this is only a minor nuisance since we can use cellular approximation to replace $f'_n$ with a function that is. The second obstacle, which I have not resolved, is that unless I am mistaken the image of $f'_n$ need not be a subcomplex of $F_n$. So when it comes to gluing $F_n$ and $CE_n'$ together I am unsure how to endow the resulting space with a CW-structure. Clearly I am just not thinking correctly, so if anyone could help over this stumbling block, that would be much appreciated.

James Miller
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  • The image of $f_n'$ shouldn't need to be a subcomplex. For example, any cellular map $\phi: S^n \to X$ gives you a new CW-complex $X \cup_\phi C(S^n)$, where $\phi$ is the attaching map for the new cell. It doesn't matter if the image is terrible. The argument for a general cellular map should roughly be the same as this: every cell of $E_n'$ gives, via $f'_n$, an attaching map for a new cell in the mapping cone. – Tyler Lawson Mar 08 '14 at 16:50
  • Ah! I see now. If you write that as an answer to the question I'd be more than happy to accept it. – James Miller Mar 08 '14 at 21:42

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