Special and general types Special notions Variants Extra structure Operations Theorems The Atiyah-Hirzebruch spectral sequence (AHSS) is a type of spectral sequence that generalizes the Serre spectral sequence from ordinary cohomology to any generalized (Eilenberg-Steenrod) cohomology theory . For any (finite) homotopy fiber sequence then the corresponding -Atiyah-Hirzebruch spectral sequence has on its second page the ordinary cohomology of with coefficients in the -cohomology groups of the fiber and converges to the proper -cohomology of the total space: This is of interest already for , as then it expresses generalized cohomology in terms of ordinary cohomology with coefficients in the base cohomology ring. (note on terminology) Often the terminology “Atiyah-Hirzebruch spectral sequence” is taken to refer to only this case with , while the general case is then referred to as “Serre spectral sequence for generalized cohomology” or similar. In (Atiyah-Hirzebruch 61,p. 17) the case is labeled “Theorem”, while the general case, stated right after the theorem, is labeled “2.2 Remark”. The proof of the theorem that is given is very short, it just says that since topological K-theory satisfies the exactness axiom of a generalized cohomology theory, it is immediate that the conditions for a spectral sequence stated as Axioms (SP.1)-(SP.5) in (Cartan-Eilenberg 56, section XV.7) are met. Indeed Example 2 in (Cartan-Eilenberg 56, section XV.7) observes that the spectral sequence in question exists for “some fixed cohomology theory” because “Axioms (SP.1)-(SP.4) are consequences of usual properties of cohomology groups”. In view of this, the contribution of (Atiyah-Hirzebruch 61) would not be so much the observation of what is now called the AHSS, rather than the proof that K-theory satisfies the axioms of a generalized cohomology theory. Indeed, according to (Adams 74, p. 127-128, 215), the AHSS was earlier observed by George Whitehead and “then became a folk-theorem”...