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!