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 ${\displaystyle X}$ and a generalized cohomology theory ${\displaystyle E^{\bullet }}$, it relates the generalized cohomology groups

${\displaystyle E^{i}(X)}$

with 'ordinary' cohomology groups ${\displaystyle H^{j}}$ with coefficients in the generalized cohomology of a point. More precisely, the ${\displaystyle E_{2}}$ term of the spectral sequence is ${\displaystyle H^{p}(X;E^{q}(pt))}$, and the spectral sequence converges conditionally to ${\displaystyle E^{p+q}(X)}$.

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 ${\displaystyle E=H_{\text{Sing}}}$. It can be derived from an exact couple that gives the ${\displaystyle E_{1}}$ page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with ${\displaystyle E}$. In detail, assume ${\displaystyle X}$ to be the total space of a Serre fibration with fibre ${\displaystyle F}$ and base space ${\displaystyle B}$. The filtration of ${\displaystyle B}$ by its ${\displaystyle n}$-skeletons ${\displaystyle B_{n}}$ gives rise to a filtration of ${\displaystyle X}$. There is a corresponding spectral sequence with ${\displaystyle E_{2}}$ term

${\displaystyle H^{p}(B;E^{q}(F))}$

and converging to the associated graded ring of the filtered ring

${\displaystyle E_{\infty }^{p,q}=E^{p+q}(X)}$.

This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre ${\displaystyle F}$ is a point.