Atiyah–Hirzebruch spectral sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by Michael Atiyah and Friedrich Hirzebruch (1961) in the special case of topological K-theory. For a CW complex and a generalized cohomology theory , it relates the generalized cohomology groups
with 'ordinary' cohomology groups with coefficients in the generalized cohomology of a point. More precisely, the term of the spectral sequence is , and the spectral sequence converges conditionally to .
Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where . It can be derived from an exact couple that gives the page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with . In detail, assume to be the total space of a Serre fibration with fibre and base space . The filtration of by its -skeletons gives rise to a filtration of . There is a corresponding spectral sequence with term
and abutting to the associated graded ring of the filtered ring
This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre is a point.
For example, the complex topological -theory of a point is
- where is in degree
This implies that the terms on the -page of a finite CW-complex looks like
Since the -theory of a point is
we can always guarantee that
This implies that the spectral sequence collapses on for many spaces. This can be checked on every , algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in .
Cotangent Bundle on a Circle
For example, consider the cotangent bundle . This is a fiber bundle with fiber so the -page reads as
The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For it is the Steenrod square where we take it as the composition
where is reduction mod and is the Bockstein homomorphism (connecting morphism) from the short exact sequence
Complete Intersection 3-fold
Consider a smooth complete intersection 3-fold (such as a complete intersection Calabi-Yau 3-fold). If we look at the -page of the spectral sequence
we can see immediately that the only potentially non-trivial differentials are
It turns out that these differentials vanish in both cases, hence . In the first case, since is trivial for we have the first set of differentials are zero. The second set are trivial because sends the identification shows the differential is trivial.
The Atiyah–Hirzebruch spectral sequence can be used to compute twisted K-theory groups as well. In short, twisted K-theory is the group completion of the isomorphism classes of vector bundles defined by gluing data where
for some cohomology class . Then, the spectral sequence reads as
but with different differentials. For example,
On the -page the differential is
Higher odd-dimensional differentials are given by Massey products for twisted K-theory tensored by . So
Note that if the underlying space is formal, meaning its rational homotopy type is determined by its rational cohomology, hence has vanishing Massey products, then the odd-dimensional differentials are zero. Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan proved this for all compact Kähler manifolds, hence in this case. In particular, this includes all smooth projective varieties.
Twisted K-theory of 3-Sphere
The twisted K-theory for can be readily computed. First of all, since and , we have that the differential on the -page is just cupping with the class given by . This gives the computation
Recall that the rational bordism group is isomorphic to the ring
generated by the bordism classes of the (complex) even dimensional projective spaces in degree . This gives a computationally tractable spectral sequence for computing the rational bordism groups.
Recall that where . Then, we can use this to compute the complex cobordism of a space via the spectral sequence. We have the -page given by
- Davis, James; Kirk, Paul, Lecture Notes in Algebraic Topology (PDF)
- Atiyah, Michael Francis; Hirzebruch, Friedrich (1961), "Vector bundles and homogeneous spaces", Proc. Sympos. Pure Math., Vol. III, Providence, R.I.: American Mathematical Society, pp. 7–38, MR 0139181
- Atiyah, Michael, Twisted K-Theory and cohomology, arXiv:math/0510674, Bibcode:2005math.....10674A