In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, it relates the generalized cohomology groups with 'ordinary' cohomology groups H^j with coefficients in the generalized cohomology of a point. More precisely, the E_2 term of the spectral sequence is, and the spectral sequence converges conditionally to E^{p+q}(X). 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 E_1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E. In detail, assume X to be the total space of a Serre fibration with fibre F and base space B. The filtration of B by its n-skeletons B_n gives rise to a filtration of X. There is a corresponding spectral sequence with E_2 term and converging to the associated graded ring of the filtered ring This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre F is a point.

Examples

Topological K-theory

For example, the complex topological K-theory of a point is By definition, the terms on the E_2-page of a finite CW-complex X look like Since the K-theory of a point is we can always guarantee that This implies that the spectral sequence collapses on E_2 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 of S^1. This is a fiber bundle with fiber \mathbb{R} so the E_2-page reads as

Differentials

The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For d_3 it is the Steenrod square Sq^3 where we take it as the composition where r is reduction mod 2 and \beta is the Bockstein homomorphism (connecting morphism) from the short exact sequence

Complete intersection 3-fold

Consider a smooth complete intersection 3-fold X (such as a complete intersection Calabi-Yau 3-fold). If we look at the E_2-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 k > i we have the first set of differentials are zero. The second set are trivial because Sq^2 sends the identification shows the differential is trivial.

Twisted K-theory

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 E_3-page the differential is Higher odd-dimensional differentials d_{2k+1} are given by Massey products for twisted K-theory tensored by \mathbb{R}. 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 S^3 can be readily computed. First of all, since and, we have that the differential on the E_3-page is just cupping with the class given by \lambda. This gives the computation

Rational bordism

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 4k. This gives a computationally tractable spectral sequence for computing the rational bordism groups.

Complex cobordism

Recall that where. Then, we can use this to compute the complex cobordism of a space X via the spectral sequence. We have the E_2-page given by