Atiyah–Hirzebruch spectral sequence

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In mathematics, the Atiyah–Hirzebruch spectral sequence is a computational tool from homological algebra, designed to make possible the calculation of an extraordinary cohomology theory. For a CW complex X, or more general topological space, it puts in relation the extraordinary cohomology groups

h ⁱ(X)

with 'ordinary' cohomology groups (such as singular cohomology H^j) with various coefficient groups. The hallmark of extraordinary theories is that h(.) applied to a point has non-zero values in dimensions other than dimension zero.

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

h^{p + q}(X).

This can yield computational information even in the case where the fibre F is a point.

[edit] Applications

The original construction of the spectral sequence, by Michael Atiyah and Friedrich Hirzebruch, was for K-theory. It was later applied more broadly, to other cohomology theories.

The Atiyah-Hirzebruch spectral sequence is now used widely in theoretical physics: see K-theory (physics).

[edit] References

• M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces (1961) Amer. Math. Soc. Symp. in Pure Math. III(1961) 7–38.