# nLab Atiyah–Hirzebruch spectral sequence

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

cohomology

# Contents

## Idea

The Atiyah-Hirzebruch spectral sequence is a kind of spectral sequence that generalizes the Serre spectral sequence from ordinary cohomology $H^\bullet$ to any generalized (Eilenberg-Steenrod) cohomology theory $E^\bullet$.

For any homotopy fiber sequence

$\array{ F &\longrightarrow& P \\ && \downarrow \\ && X }$

then the coresponding $E$-Atiyah-Hirzebruch spectral sequence has on its second page the ordinary cohomology of $X$ with coefficients in the $E$-cohomology groups of the fiber and coverges to the proper $E$-cohomology of the total space:

$E_2^{p,q} = H^p(X,E^q(F)) \Rightarrow E^{\bullet}(P) \,.$

This is of interest already for $F \simeq \ast$, as then it expresses generalized cohomology in terms of cordinary cohomology with coefficients in the base cohomology ring.

## Examples

### For K-theory

For $E =$ KU, hence for K-theory the differential $d_3$ of the Atiyah-Hirzebruch spectral sequence with $F = \ast$ is given by a Steenrod square operation

$d_3 = Sq^3$

For twisted K-theory this picks up in addition the cup product with the 3-class $H$ of the twist:

$d_3 = Sq^3 + H \cup(-)$

(Rosenberg 82, Atiyah-Segal 05 (4.1)). The higher differentials $d_5$ and $d_7$ here are given by higher Massey products with the twisting class (Atiyah-Segal 05 sections 5-7).

## Applications

### D-brane charges in string theory

In string theory D-brane charges are classes in $E = KU$-cohomology, i.e. in K-theory. The second page of of the corresponding Atiyah-Hirzebruch spectral sequence (see above) for $F = \ast$ hence expresses ordinary cohomology in all even or all odd degrees, and being in the kernel of all the differentials is hence the constraint on such ordinary cohomology data to lift to genuine K-theory classes, hence to genuine D-brane charges. In this way the Atiyah-Hirzebruch spectral sequences is used in (Maldacena-Moore-Seiberg 01, Evslin-Sati 06)

## References

The original article with application to K-theory is

• M. F. Atiyah, F. Hirzebruch, Vector bundles and homogeneous spaces, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I., MR 0139181

and further discussion of the case of twisted K-theory is due to

Application for the case of K-theory to D-brane charges in string theory is discussed in

and detailed review of this is in

Discussion for the case of Morava K-theory and Morava E-theory with comments on application to charges of M-branes is in

Application to motivic cobordism cohomology theory is discussed in

• Nobuaki Yagita, Applications of the Atiyah-Hirzebruch spectral sequences for motivic cobordisms (pdf)

Revised on May 11, 2015 21:31:27 by Urs Schreiber (89.204.137.99)