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 and a generalized cohomology theory , it relates the generalized cohomology groups

with 'ordinary' cohomology groups with coefficients in the generalized cohomology of a point. More precisely, the term of the spectral sequence is , and the spectral sequence converges conditionally to .

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 . It can be derived from an exact couple that gives the page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with . In detail, assume to be the total space of a Serre fibration with fibre and base space . The filtration of by its -skeletons gives rise to a filtration of . There is a corresponding spectral sequence with term

and converging to the associated graded ring of the filtered ring

.

This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre is a point.

Examples

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Topological K-theory

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For example, the complex topological  -theory of a point is

  where   is in degree  

By definition, the terms on the  -page of a finite CW-complex   look like

 

Since the  -theory of a point is

 

we can always guarantee that

 

This implies that the spectral sequence collapses on   for many spaces. This can be checked on every  , algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in  .

Cotangent bundle on a circle

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For example, consider the cotangent bundle of  . This is a fiber bundle with fiber   so the  -page reads as

 

Differentials

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The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For   it is the Steenrod square   where we take it as the composition

 

where   is reduction mod   and   is the Bockstein homomorphism (connecting morphism) from the short exact sequence

 

Complete intersection 3-fold

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Consider a smooth complete intersection 3-fold   (such as a complete intersection Calabi-Yau 3-fold). If we look at the  -page of the spectral sequence

 

we can see immediately that the only potentially non-trivial differentials are

 

It turns out that these differentials vanish in both cases, hence  . In the first case, since   is trivial for   we have the first set of differentials are zero. The second set are trivial because   sends   the identification   shows the differential is trivial.

Twisted K-theory

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The Atiyah–Hirzebruch spectral sequence can be used to compute twisted K-theory groups as well. In short, twisted K-theory is the group completion of the isomorphism classes of vector bundles defined by gluing data   where

 

for some cohomology class  . Then, the spectral sequence reads as

 

but with different differentials. For example,

 

On the  -page the differential is

 

Higher odd-dimensional differentials   are given by Massey products for twisted K-theory tensored by  . So

 

Note that if the underlying space is formal, meaning its rational homotopy type is determined by its rational cohomology, hence has vanishing Massey products, then the odd-dimensional differentials are zero. Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan proved this for all compact Kähler manifolds, hence   in this case. In particular, this includes all smooth projective varieties.

Twisted K-theory of 3-sphere

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The twisted K-theory for   can be readily computed. First of all, since   and  , we have that the differential on the  -page is just cupping with the class given by  . This gives the computation

 

Rational bordism

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Recall that the rational bordism group   is isomorphic to the ring

 

generated by the bordism classes of the (complex) even dimensional projective spaces   in degree  . This gives a computationally tractable spectral sequence for computing the rational bordism groups.

Complex cobordism

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Recall that   where  . Then, we can use this to compute the complex cobordism of a space   via the spectral sequence. We have the  -page given by

 

See also

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References

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  • Davis, James; Kirk, Paul, Lecture Notes in Algebraic Topology (PDF), archived from the original (PDF) on 2016-03-04, retrieved 2017-08-12
  • Atiyah, Michael Francis; Hirzebruch, Friedrich (1961), "Vector bundles and homogeneous spaces", Proc. Sympos. Pure Math., Vol. III, Providence, R.I.: American Mathematical Society, pp. 7–38, MR 0139181
  • Atiyah, Michael, Twisted K-Theory and cohomology, arXiv:math/0510674, Bibcode:2005math.....10674A