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

${\displaystyle E^{i}(X)}$

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

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 ${\displaystyle E=H_{\text{Sing}}}$. It can be derived from an exact couple that gives the ${\displaystyle E_{1}}$ page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with ${\displaystyle E}$. In detail, assume ${\displaystyle X}$ to be the total space of a Serre fibration with fibre ${\displaystyle F}$ and base space ${\displaystyle B}$. The filtration of ${\displaystyle B}$ by its ${\displaystyle n}$-skeletons ${\displaystyle X_{n}}$ gives rise to a filtration of ${\displaystyle X}$. There is a corresponding spectral sequence with ${\displaystyle E_{2}}$ term

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

and abutting to the associated graded ring of the filtered ring

${\displaystyle E_{2}^{p,q}=E^{p+q}(X)}$.

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

Examples

Topological K-theory

For example, the complex topological ${\displaystyle K}$-theory of a point is

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

This implies that the terms on the ${\displaystyle E_{2}}$-page of a finite CW-complex ${\displaystyle X}$ looks like

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

Since the ${\displaystyle K}$-theory of a point is

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

we can always guarantee that

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

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

Cotangent Bundle on a Circle

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

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

Differentials

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

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

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

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

Complete Intersection 3-fold

Consider a smooth complete intersection 3-fold ${\displaystyle X}$ (such as a complete intersection Calabi-Yau 3-fold). If we look at the ${\displaystyle E_{2}}$-page of the spectral sequence

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

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

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

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

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

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

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

but with different differentials. For example,

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

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

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

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

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

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

Twisted K-theory of 3-Sphere

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

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

Rational Bordism

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

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

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

Complex Cobordism

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

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

References

• Davis, James; Kirk, Paul, Lecture Notes in Algebraic Topology (PDF)
• 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