Atiyah–Hirzebruch_spectral_sequence

Atiyah–Hirzebruch spectral sequence

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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}=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))

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