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 .
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 cordinary 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” which was “eventually published by Atiyah and Hirzebruch”.
Maybe it should be called the “Cartan-Eilenberg-Whitehead spectral sequence”.
For genuine G-equivariance, with RO(G)-grading for representation spheres , then for an -fibration of topological G-spaces and for any -Mackey functor, the equivariant Serre spectral sequence looks like (Kronholm 10, theorem 3.1):
Let be a an additive unreduced generalized cohomology functor (def.). Let be a CW-complex and let be a Serre fibration, such that all its fibers are weakly contractible or such that is simply connected. In either case all fibers are identified with a typical fiber up to weak homotopy equivalence by connectedness (this example), and well defined up to unique iso in the homotopy category by simply connectedness:
If at least one of the following two conditions is met
then there is a cohomology spectral sequence, (def.), whose -page is the ordinary cohomology of with coefficients in the -cohomology groups of the fiber, and which converges to the -cohomology groups of the total space
with respect to the filtering given by
where is the fiber over the th stage of the CW-complex .
Generally, without assumptions on the connectivity of , there is a spectral sequence of this form with ordinary cohomology with coefficients in replaced by ordinary cohomology with local coefficients .
where we take and .
In order to determine the -page, we analyze the -page: By definition
Let be the set of -dimensional cells of , and notice that for then
This implies that
Since cellular cohomology of a CW-complex agrees with its singular cohomology (thm.), hence with its ordinary cohomology, to conclude that the -page is as claimed, it is now sufficient to show that the differential coincides with the differential in the cellular cochain complex (def.).
We discuss this now for , hence and . (The general case works the same, just with various factors of replacing the point.)
Here the bottom vertical morphisms are those induced from any chosen cell inclusion .
The differential in the spectral sequence is the middle horizontal composite. From this the vertical isomorphisms give the top horizontal map. But the bottom horizontal map identifies this top horizontal morphism componentwise with the restriction to the boundary of cells. Hence the top horizontal morphism is indeed the coboundary operator for the cellular cohomology of with coefficients in (def.). This cellular cohomology coincides with singular cohomology of the CW-complex (thm.), hence computes the ordinary cohomology of .
Now to see the convergence. If is finite dimensional then the convergence condition as stated in this prop. is met. Alternatively, if is bounded below in degree, then by the above analysis the -page has a horizontal line below which it vanishes. Accordingly the same is then true for all higher pages, by each of them being the cohomology of the previous page. Since the differentials go right and down, eventually they pass beneath this vanishing line and become 0. This is again the condition needed in the proof of this prop. to obtain convergence.
By that proposition the convergence is to the inverse limit
One also gets the Atiyah-Hirzebruch spectral sequence, up to isomorphism, by instead using the filtering given by the Postnikov tower of an Omega-spectrum representing the given generalized cohomology theory.
This is due to (Maunder 63, theorem 3.3)
|tower diagram/filtering||spectral sequence of a filtered stable homotopy type|
|filtered chain complex||spectral sequence of a filtered complex|
|Postnikov tower||Atiyah-Hirzebruch spectral sequence|
|chromatic tower||chromatic spectral sequence|
|skeleta of simplicial object||spectral sequence of a simplicial stable homotopy type|
|skeleta of Sweedler coring of E-∞ algebra||Adams spectral sequence|
|filtration by support||…|
|slice filtration||slice spectral sequence|
for a ring spectrum, then the AHSS is multiplicative…
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)
The statement of the existence of the spectral sequence first appears in print (with topological K-theory) in
but the proof given consists essentially in pointing to section XV.7 (“a more general setting in which the theory of spectral sequences may be developed”) of
This is by filtering over the stages of the base space CW-complex. That one gets an isomorphic spectral sequence by instead filtering over the Postnikov tower of any Omega-spectrum representing the given generalized cohomology theory is due to
Early lecture notes include
where the idea is attributed to George Whitehead:
the Atiyah-Hirzebruch spectral sequence, which was really invented by G. W. Whitehead but not published by him (Adams 74, p. 127-128)
These spectral sequences were probably first invented by G. W. Whitehead, but he got them just after he wrote the paper Whitehead 56 ‘Homotopy groups of joins and unions’ in which they ought to have appeared. They then became a folk-theorem and were eventually published by Atiyah and Hirzebruch (Adams 74, p. 215)
A more detailed account of the proof is in
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.
Discussion in genuine equivariant cohomology, i.e. including RO(G)-grading, is in
and detailed review of this is in