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.
and a generalized cohomology theory
with coefficients in the generalized cohomology of a point.
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
to be the total space of a Serre fibration with fibre
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
-theory of a point is By definition, the terms on the
-theory of a point is we can always guarantee that This implies that the spectral sequence collapses on
, algebraic curves, or spaces with non-zero cohomology in even degrees.
Therefore, it collapses for all (complex) even dimensional smooth complete intersections in
-page reads as The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed.
is the Bockstein homomorphism (connecting morphism) from the short exact sequence Consider a smooth complete intersection 3-fold
(such as a complete intersection Calabi-Yau 3-fold).
-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
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
Then, the spectral sequence reads as but with different 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 particular, this includes all smooth projective varieties.
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
This gives a computationally tractable spectral sequence for computing the rational bordism groups.
Then, we can use this to compute the complex cobordism of a space