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 $E^(X)$.

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 $E=H_$. 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

:$E_\infty^ = E^(X)$.

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 ,x^/math> where $x$ is in degree $2$
By definition, the terms on the $E_2$-page of a finite CW-complex $X$ look like
:$E_2^(X) = H^p(X;KU^q(pt))$
Since the $K$-theory of a point is
:$K^q(pt) = \begin \mathbb & \text \\ 0 & \text \end$
we can always guarantee that
:$E_2^(X) = 0$
This implies that the spectral sequence collapses on $E_2$ for many spaces. This can be checked on every $\mathbb^n$, algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in $\mathbb^n$.

Cotangent bundle on a circle

For example, consider the cotangent bundle of $S^1$. This is a fiber bundle with fiber $\mathbb$ so the $E_2$-page reads as
:$\begin \vdots &\vdots & \vdots \\ 2 & H^0(S^1;\mathbb) & H^1(S^1;\mathbb) \\ 1 & 0 & 0 \\ 0 & H^0(S^1;\mathbb) & H^1(S^1;\mathbb) \\ -1 & 0 & 0 \\ -2 & H^0(S^1;\mathbb) & H^1(S^1;\mathbb) \\ \vdots &\vdots & \vdots \\ \hline & 0 & 1 \end$

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
:$0 \to \mathbb \to \mathbb \to \mathbb/2 \to 0$

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
:$\begin \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 2 & H^0(X; \mathbb) & 0 & H^2(X;\mathbb) & H^3(X;\mathbb) & H^4(X;\mathbb) & 0 & H^6(X;\mathbb) \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & H^0(X; \mathbb) & 0 & H^2(X;\mathbb) & H^3(X;\mathbb) & H^4(X;\mathbb) & 0 & H^6(X;\mathbb)\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -2 & H^0(X; \mathbb) & 0 & H^2(X;\mathbb) & H^3(X;\mathbb) & H^4(X;\mathbb) & 0 & H^6(X;\mathbb)\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \end$
we can see immediately that the only potentially non-trivial differentials are
:$\begin d_3:E_3^ \to E_3^ \\ d_3:E_3^ \to E_3^ \end$
It turns out that these differentials vanish in both cases, hence $E_2 = E_\infty$. In the first case, since $Sq^k:H^i(X;\mathbb/2) \to H^(X;\mathbb/2)$ is trivial for $k > i$ we have the first set of differentials are zero. The second set are trivial because $Sq^2$ sends $H^3(X;\mathbb/2) \to H^5(X) = 0$ 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 $(U_,g_)$ where
:$g_g_g_ = \lambda_$
for some cohomology class $\lambda \in H^3(X,\mathbb)$. Then, the spectral sequence reads as
:$E_2^ = H^p(X;KU^q(*)) \Rightarrow KU^_\lambda(X)$
but with different differentials. For example,
:$E_3^ = E_2^ = \begin \vdots & \vdots & \vdots & \vdots & \vdots \\ 2 & H^0(S^3;\mathbb) & 0 & 0 & H^3(S^3;\mathbb) \\ 1 & 0 & 0 & 0 & 0 \\ 0 & H^0(S^3;\mathbb) & 0 & 0 & H^3(S^3;\mathbb) \\ -1 & 0 & 0 & 0 & 0 \\ -2 & H^0(S^3;\mathbb) & 0 & 0 & H^3(S^3;\mathbb) \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline& 0 & 1 & 2 & 3 \end$
On the $E_3$-page the differential is
:$d_3 = Sq^3 + \lambda$
Higher odd-dimensional differentials $d_$ are given by Massey products for twisted K-theory tensored by $\mathbb$. So
:$\begin d_5 &= \ \\ d_7 &= \ \end$
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
:$KU_\lambda^k = \begin \mathbb & k \text \\ \mathbb/\lambda & k \text \end$

Rational bordism

Recall that the rational bordism group $\Omega_*^\otimes \mathbb$ is isomorphic to the ring
: \mathbb \mathbb^0 mathbb^2 mathbb^4 mathbb^6\ldots]
generated by the bordism classes of the (complex) even dimensional projective spaces mathbb^/math> in degree $4k$. This gives a computationally tractable spectral sequence for computing the rational bordism groups.

Complex cobordism

Recall that MU^*(pt) = \mathbb _1,x_2,\ldots/math> where $x_i \in \pi_(MU)$. 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
:$E_2^ = H^p(X;MU^q(pt))$

See also

* Quillen–Lichtenbaum conjecture

References

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{{DEFAULTSORT:Atiyah-Hirzebruch spectral sequence
Spectral sequences
K-theory