Atiyah–Hirzebruch spectral sequence
- Atiyah–Hirzebruch spectral sequence
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 "E"2 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.
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).
References
* M. F. Atiyah and F. Hirzebruch, "Vector bundles and homogeneous spaces" (1961) Amer. Math. Soc. Symp. in Pure Math. III(1961) 7–38.
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