Atiyah–Hirzebruch spectral sequence

In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by 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.

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{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} $$ It turns out that these differentials vanish in both cases, hence $$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
 * $$ E_2^{p,q} = H^p(X;KU^q(*)) \Rightarrow KU^{p+q}_\lambda(X)$$

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{align} d_5 &= \{ \lambda, \lambda, - \} \\ d_7 &= \{ \lambda, \lambda, \lambda, - \} \end{align} $$ Note that if the underlying space is formal, meaning its rational homotopy type is determined by its rational cohomology, hence has vanishing Massey products, then the odd-dimensional differentials are zero. Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan proved this for all compact Kähler manifolds, hence $$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
 * $$E_2^{p,q} = H^p(X;MU^q(pt))$$