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Atiyah–Hirzebruch spectral sequence

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Idea

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

For any homotopy fiber sequence

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

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

E 2 p,q=H p(X,E q(F))E (P). E_2^{p,q} = H^p(X,E^q(F)) \Rightarrow E^{\bullet}(P) \,.

This is of interest already for F*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=E = KU, hence for K-theory the differential d 3d_3 of the Atiyah-Hirzebruch spectral sequence with F=*F = \ast is given by a Steenrod square operation

d 3=Sq 3 d_3 = Sq^3

(Atiyah-Hirzebruch 61).

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

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

(Rosenberg 82, Atiyah-Segal 05 (4.1)). The higher differentials d 5d_5 and d 7d_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=KUE = KU-cohomology, i.e. in K-theory. The second page of of the corresponding Atiyah-Hirzebruch spectral sequence (see above) for F=*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)

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Whitehead towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Cech nerve of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

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)