The Atiyah-Hirzebruch spectral sequence is a kind of spectral sequence that generalizes the Serre spectral sequence from ordinary cohomology to any generalized (Eilenberg-Steenrod) cohomology theory .
For any homotopy fiber sequence
then the coresponding -Atiyah-Hirzebruch spectral sequence has on its second page the ordinary cohomology of with coefficients in the -cohomology groups of the fiber and coverges to the proper -cohomology of the total space:
This is of interest already for , as then it expresses generalized cohomology in terms of cordinary cohomology with coefficients in the base cohomology ring.
In string theory D-brane charges are classes in -cohomology, i.e. in K-theory. The second page of of the corresponding Atiyah-Hirzebruch spectral sequence (see above) for hence expresses ordinary cohomology in all even or all odd degrees, and being in the kernel of all the differentials is hence the constraint on such ordinary cohomology data to lift to genuine K-theory classes, hence to genuine D-brane charges. In this way the Atiyah-Hirzebruch spectral sequences is used in (Maldacena-Moore-Seiberg 01, Evslin-Sati 06)
|tower diagram/filtering||spectral sequence of a filtered stable homotopy type|
|filtered chain complex||spectral sequence of a filtered complex|
|Whitehead tower||Atiyah-Hirzebruch spectral sequence|
|chromatic tower||chromatic spectral sequence|
|skeleta of simplicial object||spectral sequence of a simplicial stable homotopy type|
|skeleta of Cech nerve of E-∞ algebra||Adams spectral sequence|
|filtration by support||…|
|slice filtration||slice spectral sequence|
The original article with application to K-theory is
and further discussion of the case of twisted K-theory is due to
Jonathan Rosenberg, Homological Invariants of Extensions of -algebras, Proc. Symp. Pure Math 38 (1982) 35.
and detailed review of this is in