Excision in the stable homotopy category Is there a way to make precise the statements (if it holds at all) that excision holds in the stable homotopy category? I am a beginner in this kind of things, and the stable homotopy category for me is just CW spectra with maps up to homotopy, following Adams.
I mean, suppose you have an inclusion of subspaces $B \subset A$ of a topological space $X$ with $\overline B \subset A°$. Is it true that $(X-B, A-B)$ is stably homotopically equivalent to $(X, A)$? You may make assumptions on these spaces, if needed.
I have the vague feeling that this is true since $\Omega$-spectra correspond to cohomology theories, for which exchision hold, but I can not make this statement precise or prove it.
 A: The only reasonable notion of stable homotopy equivalence of pairs $(A,B) \rightarrow (C,D)$ that is compatible with your question is an equivalence $\Sigma^\infty (A \cup \operatorname{cone} (B)) \rightarrow \Sigma^\infty (C \cup \operatorname{cone}(D))$.
Now given a map of pairs $(A,B) \rightarrow (C,D)$ that induces an isomorphism on relative homology, it is easy to show (provided you know basic stable homotopy theory) that the induced map $\Sigma^\infty (A \cup \operatorname{cone} (B)) \rightarrow \Sigma^\infty (C \cup \operatorname{cone}(D))$ is a stable equivalence. This is because relative homology coincides with homology of the mapping cone of the inclusion (this is a consequence of excision), and a map of finite spectra is an equivalence, if and only if, it is a homology equivalence (this is Whitehead's theorem in spectra).
Hence, the homological statement of excision implies the stable homotopy version of excision, i.e.  $\Sigma^\infty (X - B \cup \operatorname{cone} (A-B)) \rightarrow \Sigma^\infty (X \cup \operatorname{cone}(A))$ is an equivalence.
