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 $X$ and a generalized cohomology theory $E^{\bullet }$, it relates the generalized cohomology groups

$E^{i}(X)$

with 'ordinary' cohomology groups $H^{j}$ with coefficients in the generalized cohomology of a point. More precisely, the $E_{2}$ term of the spectral sequence is $H^{p}(X;E^{q}(pt))$, and the spectral sequence converges conditionally to $E^{p+q}(X)$.

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 $E=H_{\text{Sing}}$. It can be derived from an exact couple that gives the $E_{1}$ 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 $B_{n}$ gives rise to a filtration of $X$. There is a corresponding spectral sequence with $E_{2}$ term

$H^{p}(B;E^{q}(F))$

and converging to the associated graded ring of the filtered ring

$E_{\infty }^{p,q}\Rightarrow E^{p+q}(X)$.

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

Examples

Topological K-theory

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

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

By definition, the terms on the $E_{2}$-page of a finite CW-complex $X$ look like

$E_{2}^{p,q}(X)=H^{p}(X;KU^{q}(pt))$

Since the $K$-theory of a point is

$K^{q}(pt)={\begin{cases}\mathbb {Z} &{\text{if q is even}}\\0&{\text{otherwise}}\end{cases}}$

we can always guarantee that

$E_{2}^{p,2k+1}(X)=0$

This implies that the spectral sequence collapses on $E_{2}$ for many spaces. This can be checked on every $\mathbb {CP} ^{n}$, algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in $\mathbb {CP} ^{n}$.

Cotangent bundle on a circle

For example, consider the cotangent bundle of $S^{1}$. This is a fiber bundle with fiber $\mathbb {R}$ so the $E_{2}$-page reads as

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

Differentials

The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For $d_{3}$ it is the Steenrod square $Sq^{3}$ where we take it as the composition

$\beta \circ Sq^{2}\circ r$

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

$0\to \mathbb {Z} \to \mathbb {Z} \to \mathbb {Z} /2\to 0$

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 $E_{2}$-page of the spectral sequence

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

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

${\begin{aligned}d_{3}:E_{3}^{0,2k}\to E_{3}^{3,2k-2}\\d_{3}:E_{3}^{3,2k}\to E_{3}^{6,2k-2}\end{aligned}}$

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

$g_{ij}g_{jk}g_{ki}=\lambda _{ijk}$

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

$E_{2}^{p,q}=H^{p}(X;KU^{q}(*))\Rightarrow KU_{\lambda }^{p+q}(X)$

but with different differentials. For example,

$E_{3}^{p,q}=E_{2}^{p,q}={\begin{array}{c|cccc}\vdots &\vdots &\vdots &\vdots &\vdots \\2&H^{0}(S^{3};\mathbb {Z} )&0&0&H^{3}(S^{3};\mathbb {Z} )\\1&0&0&0&0\\0&H^{0}(S^{3};\mathbb {Z} )&0&0&H^{3}(S^{3};\mathbb {Z} )\\-1&0&0&0&0\\-2&H^{0}(S^{3};\mathbb {Z} )&0&0&H^{3}(S^{3};\mathbb {Z} )\\\vdots &\vdots &\vdots &\vdots &\vdots \\\hline &0&1&2&3\end{array}}$

On the $E_{3}$-page the differential is

$d_{3}=Sq^{3}+\lambda$

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

${\begin{aligned}d_{5}&=\{\lambda ,\lambda ,-\}\\d_{7}&=\{\lambda ,\lambda ,\lambda ,-\}\end{aligned}}$

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 $E_{\infty }=E_{4}$ in this case. In particular, this includes all smooth projective varieties.

Twisted K-theory of 3-sphere

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

$KU_{\lambda }^{k}={\begin{cases}\mathbb {Z} &k{\text{ is even}}\\\mathbb {Z} /\lambda &k{\text{ is odd}}\end{cases}}$

Rational bordism

Recall that the rational bordism group $\Omega _{*}^{\text{SO}}\otimes \mathbb {Q}$ is isomorphic to the ring

$\mathbb {Q} [[\mathbb {CP} ^{0}],[\mathbb {CP} ^{2}],[\mathbb {CP} ^{4}],[\mathbb {CP} ^{6}],\ldots ]$

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

Complex cobordism

Recall that $MU^{*}(pt)=\mathbb {Z} [x_{1},x_{2},\ldots ]$ where $x_{i}\in \pi _{2i}(MU)$. Then, we can use this to compute the complex cobordism of a space $X$ via the spectral sequence. We have the $E_{2}$-page given by

$E_{2}^{p,q}=H^{p}(X;MU^{q}(pt))$

See also

• Quillen–Lichtenbaum conjecture

References

• 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