Atiyah–Hirzebruch spectral sequence

In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by Atiyah & Hirzebruch (1961) in the special case of topological K-theory. For a CW complex X, it relates the generalized cohomology groups

hⁱ(X)

with 'ordinary' cohomology groups H ^j with coefficients in the generalized cohomology of a point. More precisely, the E₂ term of the spectral sequence is Hⁱ(X,h_j(point)), and the spectral sequence converges conditionally to h^i+j(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 h=H. It can be derived from an exact couple that gives the E₁ page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with h. 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 gives rise to a filtration of X. There is a corresponding spectral sequence with E₂ term

H^p(B;h ^q(F))

and abutting to the associated graded ring of the filtered ring

h^{p + q}(X).

This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre F is a point.

References

• 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