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 [math]\displaystyle{ X }[/math] and a generalized cohomology theory [math]\displaystyle{ E^\bullet }[/math], it relates the generalized cohomology groups

[math]\displaystyle{ E^i(X) }[/math]

with 'ordinary' cohomology groups [math]\displaystyle{ H^j }[/math] with coefficients in the generalized cohomology of a point. More precisely, the [math]\displaystyle{ E_2 }[/math] term of the spectral sequence is [math]\displaystyle{ H^p(X;E^q(pt)) }[/math], and the spectral sequence converges conditionally to [math]\displaystyle{ E^{p+q}(X) }[/math].

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 [math]\displaystyle{ E=H_{\text{Sing}} }[/math]. It can be derived from an exact couple that gives the [math]\displaystyle{ E_1 }[/math] page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with [math]\displaystyle{ E }[/math]. In detail, assume [math]\displaystyle{ X }[/math] to be the total space of a Serre fibration with fibre [math]\displaystyle{ F }[/math] and base space [math]\displaystyle{ B }[/math]. The filtration of [math]\displaystyle{ B }[/math] by its [math]\displaystyle{ n }[/math]-skeletons [math]\displaystyle{ B_n }[/math] gives rise to a filtration of [math]\displaystyle{ X }[/math]. There is a corresponding spectral sequence with [math]\displaystyle{ E_2 }[/math] term

[math]\displaystyle{ H^p(B; E^q(F)) }[/math]

and converging to the associated graded ring of the filtered ring

[math]\displaystyle{ E_\infty^{p,q} = E^{p+q}(X) }[/math].

This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre [math]\displaystyle{ F }[/math] is a point.

Examples

Topological K-theory

For example, the complex topological [math]\displaystyle{ K }[/math]-theory of a point is

[math]\displaystyle{ KU(*) = \mathbb{Z}[x,x^{-1}] }[/math] where [math]\displaystyle{ x }[/math] is in degree [math]\displaystyle{ 2 }[/math]

By definition, the terms on the [math]\displaystyle{ E_2 }[/math]-page of a finite CW-complex [math]\displaystyle{ X }[/math] look like

[math]\displaystyle{ E_2^{p,q}(X) = H^p(X;KU^q(pt)) }[/math]

Since the [math]\displaystyle{ K }[/math]-theory of a point is

[math]\displaystyle{ K^q(pt) = \begin{cases} \mathbb{Z} & \text{if q is even} \\ 0 & \text{otherwise} \end{cases} }[/math]

we can always guarantee that

[math]\displaystyle{ E_2^{p,2k+1}(X) = 0 }[/math]

This implies that the spectral sequence collapses on [math]\displaystyle{ E_2 }[/math] for many spaces. This can be checked on every [math]\displaystyle{ \mathbb{CP}^n }[/math], algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in [math]\displaystyle{ \mathbb{CP}^n }[/math].

Cotangent bundle on a circle

For example, consider the cotangent bundle of [math]\displaystyle{ S^1 }[/math]. This is a fiber bundle with fiber [math]\displaystyle{ \mathbb{R} }[/math] so the [math]\displaystyle{ E_2 }[/math]-page reads as

[math]\displaystyle{ \begin{array}{c|cc} \vdots &\vdots & \vdots \\ 2 & H^0(S^1;\mathbb{Q}) & H^1(S^1;\mathbb{Q}) \\ 1 & 0 & 0 \\ 0 & H^0(S^1;\mathbb{Q}) & H^1(S^1;\mathbb{Q}) \\ -1 & 0 & 0 \\ -2 & H^0(S^1;\mathbb{Q}) & H^1(S^1;\mathbb{Q}) \\ \vdots &\vdots & \vdots \\ \hline & 0 & 1 \end{array} }[/math]

Differentials

The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For [math]\displaystyle{ d_3 }[/math] it is the Steenrod square [math]\displaystyle{ Sq^3 }[/math] where we take it as the composition

[math]\displaystyle{ \beta \circ Sq^2 \circ r }[/math]

where [math]\displaystyle{ r }[/math] is reduction mod [math]\displaystyle{ 2 }[/math] and [math]\displaystyle{ \beta }[/math] is the Bockstein homomorphism (connecting morphism) from the short exact sequence

[math]\displaystyle{ 0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2 \to 0 }[/math]

Complete intersection 3-fold

Consider a smooth complete intersection 3-fold [math]\displaystyle{ X }[/math] (such as a complete intersection Calabi-Yau 3-fold). If we look at the [math]\displaystyle{ E_2 }[/math]-page of the spectral sequence

[math]\displaystyle{ \begin{array}{c|ccccc} \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 2 & H^0(X; \mathbb{Z}) & 0 & H^2(X;\mathbb{Z}) & H^3(X;\mathbb{Z}) & H^4(X;\mathbb{Z}) & 0 & H^6(X;\mathbb{Z}) \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & H^0(X; \mathbb{Z}) & 0 & H^2(X;\mathbb{Z}) & H^3(X;\mathbb{Z}) & H^4(X;\mathbb{Z}) & 0 & H^6(X;\mathbb{Z})\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -2 & H^0(X; \mathbb{Z}) & 0 & H^2(X;\mathbb{Z}) & H^3(X;\mathbb{Z}) & H^4(X;\mathbb{Z}) & 0 & H^6(X;\mathbb{Z})\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} }[/math]

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

[math]\displaystyle{ \begin{align} d_3:E_3^{0,2k} \to E_3^{3,2k-2} \\ d_3:E_3^{3,2k} \to E_3^{6,2k-2} \end{align} }[/math]

It turns out that these differentials vanish in both cases, hence [math]\displaystyle{ E_2 = E_\infty }[/math]. In the first case, since [math]\displaystyle{ Sq^k:H^i(X;\mathbb{Z}/2) \to H^{k+i}(X;\mathbb{Z}/2) }[/math] is trivial for [math]\displaystyle{ k \gt i }[/math] we have the first set of differentials are zero. The second set are trivial because [math]\displaystyle{ Sq^2 }[/math] sends [math]\displaystyle{ H^3(X;\mathbb{Z}/2) \to H^5(X) = 0 }[/math] the identification [math]\displaystyle{ Sq^3 = \beta \circ Sq^2 \circ r }[/math] 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 [math]\displaystyle{ (U_{ij},g_{ij}) }[/math] where

[math]\displaystyle{ g_{ij}g_{jk}g_{ki} = \lambda_{ijk} }[/math]

for some cohomology class [math]\displaystyle{ \lambda \in H^3(X,\mathbb{Z}) }[/math]. Then, the spectral sequence reads as

[math]\displaystyle{ E_2^{p,q} = H^p(X;KU^q(*)) \Rightarrow KU^{p+q}_\lambda(X) }[/math]

but with different differentials. For example,

[math]\displaystyle{ E_3^{p,q} = E_2^{p,q} = \begin{array}{c|cccc} \vdots & \vdots & \vdots & \vdots & \vdots \\ 2 & H^0(S^3;\mathbb{Z}) & 0 & 0 & H^3(S^3;\mathbb{Z}) \\ 1 & 0 & 0 & 0 & 0 \\ 0 & H^0(S^3;\mathbb{Z}) & 0 & 0 & H^3(S^3;\mathbb{Z}) \\ -1 & 0 & 0 & 0 & 0 \\ -2 & H^0(S^3;\mathbb{Z}) & 0 & 0 & H^3(S^3;\mathbb{Z}) \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline& 0 & 1 & 2 & 3 \end{array} }[/math]

On the [math]\displaystyle{ E_3 }[/math]-page the differential is

[math]\displaystyle{ d_3 = Sq^3 + \lambda }[/math]

Higher odd-dimensional differentials [math]\displaystyle{ d_{2k+1} }[/math] are given by Massey products for twisted K-theory tensored by [math]\displaystyle{ \mathbb{R} }[/math]. So

[math]\displaystyle{ \begin{align} d_5 &= \{ \lambda, \lambda, - \} \\ d_7 &= \{ \lambda, \lambda, \lambda, - \} \end{align} }[/math]

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 [math]\displaystyle{ E_\infty = E_4 }[/math] in this case. In particular, this includes all smooth projective varieties.

Twisted K-theory of 3-sphere

The twisted K-theory for [math]\displaystyle{ S^3 }[/math] can be readily computed. First of all, since [math]\displaystyle{ Sq^3 = \beta \circ Sq^2 \circ r }[/math] and [math]\displaystyle{ H^2(S^3) = 0 }[/math], we have that the differential on the [math]\displaystyle{ E_3 }[/math]-page is just cupping with the class given by [math]\displaystyle{ \lambda }[/math]. This gives the computation

[math]\displaystyle{ KU_\lambda^k = \begin{cases} \mathbb{Z} & k \text{ is even} \\ \mathbb{Z}/\lambda & k \text{ is odd} \end{cases} }[/math]

Rational bordism

Recall that the rational bordism group [math]\displaystyle{ \Omega_*^{\text{SO}}\otimes \mathbb{Q} }[/math] is isomorphic to the ring

[math]\displaystyle{ \mathbb{Q}[[\mathbb{CP}^0], [\mathbb{CP}^2], [\mathbb{CP}^4],[\mathbb{CP}^6],\ldots] }[/math]

generated by the bordism classes of the (complex) even dimensional projective spaces [math]\displaystyle{ [\mathbb{CP}^{2k}] }[/math] in degree [math]\displaystyle{ 4k }[/math]. This gives a computationally tractable spectral sequence for computing the rational bordism groups.

Complex cobordism

Recall that [math]\displaystyle{ MU^*(pt) = \mathbb{Z}[x_1,x_2,\ldots] }[/math] where [math]\displaystyle{ x_i \in \pi_{2i}(MU) }[/math]. Then, we can use this to compute the complex cobordism of a space [math]\displaystyle{ X }[/math] via the spectral sequence. We have the [math]\displaystyle{ E_2 }[/math]-page given by

[math]\displaystyle{ E_2^{p,q} = H^p(X;MU^q(pt)) }[/math]

References

Davis, James; Kirk, Paul, Lecture Notes in Algebraic Topology, http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf, 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, https://books.google.com/books?id=4hE7AAAAIAAJ&pg=PA197
Atiyah, Michael, Twisted K-Theory and cohomology, Bibcode: 2005math.....10674A