I've gotten myself tied in knots about this elementary derivation of a ludicrous conclusion. Appreciate a hand straightening myself out!

(1) A fibration $F\to E \to B$ of CW complexes gives rise to a distinguished triangle of suspension spectra in the stable homotopy category.

(2) An exact triangle gives rise to a long exact sequence of abelian groups under hom.

(3) By representing cohomology in spectra we should get a long exact sequence of a fibration for every cohomology theory.

But (3) is wildly false. Which step is wrong? Many thanks!


1) is just not true. It's cofiber sequences that give you distinguished triangles, exactly as you would expect. It's true that fiber and cofiber sequences agree in spectra, but it's not true that the suspension spectrum functor preserves fiber sequences. (It's a left adjoint, so the things it preserves are colimits, not limits.)


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