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)[1] in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E∙, it relates the generalized cohomology groups

Ei(X)

with 'ordinary' cohomology groups Hj with coefficients in the generalized cohomology of a point. More precisely, the E2 term of the spectral sequence is Hp(X;Eq(pt)), and the spectral sequence converges conditionally to Ep+q(X).

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 E=HSing. It can be derived from an exact couple that gives the E1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E. 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 Bn gives rise to a filtration of X. There is a corresponding spectral sequence with E2 term

Hp(B;Eq(F))

and converging to the associated graded ring of the filtered ring

E∞p,q⇒Ep+q(X).

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

## Contents

* 1 Examples
* 1.1 Topological K-theory
* 1.1.1 Cotangent bundle on a circle
* 1.1.2 Differentials
* 1.1.3 Complete intersection 3-fold
* 1.2 Twisted K-theory
* 1.2.1 Twisted K-theory of 3-sphere
* 1.3 Rational bordism
* 1.4 Complex cobordism
* 2 See also
* 3 References

## Examples

### Topological K-theory

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

KU(*)=ℤ[x,x−1] where x is in degree 2

By definition, the terms on the E2-page of a finite CW-complex X look like

E2p,q(X)=Hp(X;KUq(pt))

Since the K-theory of a point is

Kq(pt)={ℤif q is even0otherwise

we can always guarantee that

E2p,2k+1(X)=0

This implies that the spectral sequence collapses on E2 for many spaces. This can be checked on every ℂℙn, algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in ℂℙn.

#### Cotangent bundle on a circle

For example, consider the cotangent bundle of S1. This is a fiber bundle with fiber ℝ so the E2-page reads as

⋮⋮⋮2H0(S1;ℚ)H1(S1;ℚ)1000H0(S1;ℚ)H1(S1;ℚ)−100−2H0(S1;ℚ)H1(S1;ℚ)⋮⋮⋮

#### Differentials

The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For d3 it is the Steenrod square Sq3 where we take it as the composition

β∘Sq2∘r

where r is reduction mod 2 and β is the Bockstein homomorphism (connecting morphism) from the short exact sequence

0->ℤ->ℤ->ℤ/2->0

#### Complete intersection 3-fold

Consider a smooth complete intersection 3-fold X (such as a complete intersection Calabi-Yau 3-fold). If we look at the E2-page of the spectral sequence

⋮⋮⋮⋮⋮⋮⋮⋮2H0(X;ℤ)0H2(X;ℤ)H3(X;ℤ)H4(X;ℤ)0H6(X;ℤ)100000000H0(X;ℤ)0H2(X;ℤ)H3(X;ℤ)H4(X;ℤ)0H6(X;ℤ)−10000000−2H0(X;ℤ)0H2(X;ℤ)H3(X;ℤ)H4(X;ℤ)0H6(X;ℤ)⋮⋮⋮⋮⋮⋮⋮⋮

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

d3:E30,2k->E33,2k−2d3:E33,2k->E36,2k−2

It turns out that these differentials vanish in both cases, hence E2=E∞. In the first case, since Sqk:Hi(X;ℤ/2)->Hk+i(X;ℤ/2) is trivial for k>i we have the first set of differentials are zero. The second set are trivial because Sq2 sends H3(X;ℤ/2)->H5(X)=0 the identification Sq3=β∘Sq2∘r shows the differential is trivial.

### Twisted K-theory

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 (Uij,gij) where

gijgjkgki=λijk

for some cohomology class λ∈H3(X,ℤ). Then, the spectral sequence reads as

E2p,q=Hp(X;KUq(*))⇒KUλp+q(X)

but with different differentials. For example,

E3p,q=E2p,q=⋮⋮⋮⋮⋮2H0(S3;ℤ)00H3(S3;ℤ)100000H0(S3;ℤ)00H3(S3;ℤ)−10000−2H0(S3;ℤ)00H3(S3;ℤ)⋮⋮⋮⋮⋮

On the E3-page the differential is

d3=Sq3+λ

Higher odd-dimensional differentials d2k+1 are given by Massey products for twisted K-theory tensored by ℝ. So

d5={λ,λ,−}d7={λ,λ,λ,−}

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 E∞=E4 in this case. In particular, this includes all smooth projective varieties.

#### Twisted K-theory of 3-sphere

The twisted K-theory for S3 can be readily computed. First of all, since Sq3=β∘Sq2∘r and H2(S3)=0, we have that the differential on the E3-page is just cupping with the class given by λ. This gives the computation

KUλk={ℤk is evenℤ/λk is odd

### Rational bordism

Recall that the rational bordism group Ω*SO⊗ℚ is isomorphic to the ring

ℚ[[ℂℙ0],[ℂℙ2],[ℂℙ4],[ℂℙ6],…]

generated by the bordism classes of the (complex) even dimensional projective spaces [ℂℙ2k] in degree 4k. This gives a computationally tractable spectral sequence for computing the rational bordism groups.

### Complex cobordism

Recall that MU*(pt)=ℤ[x1,x2,…] where xi∈π2i(MU). Then, we can use this to compute the complex cobordism of a space X via the spectral sequence. We have the E2-page given by

E2p,q=Hp(X;MUq(pt))

## See also

* Quillen–Lichtenbaum conjecture

## References

1. ↑ 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, https://books.google.com/books?id=4hE7AAAAIAAJ&pg=PA197

* Davis, James; Kirk, Paul, Lecture Notes in Algebraic Topology, http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf, retrieved 2017-08-12
* Atiyah, Michael, Twisted K-Theory and cohomology, Bibcode: 2005math.....10674A

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