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

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

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

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

By definition, the terms on the $E_2$-page of a finite CW-complex $X$ look like

$E_2^{p,q}(X)=H^p(X;K{U}^q(pt))$

Since the $K$-theory of a point is

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

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 ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^n$, algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^n$.

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

$\begin{array}{ccc}⋮& ⋮& ⋮\\ 2& H^0(S^1;\mathbb{\mathbb{Q}})& H^1(S^1;\mathbb{\mathbb{Q}})\\ 1& 0& 0\\ 0& H^0(S^1;\mathbb{\mathbb{Q}})& H^1(S^1;\mathbb{\mathbb{Q}})\\ -1& 0& 0\\ -2& H^0(S^1;\mathbb{\mathbb{Q}})& H^1(S^1;\mathbb{\mathbb{Q}})\\ ⋮& ⋮& ⋮\end{array}$

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 $S{q}^3$ where we take it as the composition

$\beta \circ S{q}^2\circ r$

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

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

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

$\begin{array}{cccccccc}⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 2& H^0(X;\mathbb{\mathbb{Z}})& 0& H^2(X;\mathbb{\mathbb{Z}})& H^3(X;\mathbb{\mathbb{Z}})& H^4(X;\mathbb{\mathbb{Z}})& 0& H^6(X;\mathbb{\mathbb{Z}})\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& H^0(X;\mathbb{\mathbb{Z}})& 0& H^2(X;\mathbb{\mathbb{Z}})& H^3(X;\mathbb{\mathbb{Z}})& H^4(X;\mathbb{\mathbb{Z}})& 0& H^6(X;\mathbb{\mathbb{Z}})\\ -1& 0& 0& 0& 0& 0& 0& 0\\ -2& H^0(X;\mathbb{\mathbb{Z}})& 0& H^2(X;\mathbb{\mathbb{Z}})& H^3(X;\mathbb{\mathbb{Z}})& H^4(X;\mathbb{\mathbb{Z}})& 0& H^6(X;\mathbb{\mathbb{Z}})\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\end{array}$

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

$\begin{array}{r}{d}_3:E_3^{0,2k}\to E_3^{3,2k-2}\\ d_3:E_3^{3,2k}\to E_3^{6,2k-2}\end{array}$

It turns out that these differentials vanish in both cases, hence $E_2=E_{\mathrm{\infty }}$. In the first case, since $S{q}^k:H^i(X;\mathbb{\mathbb{Z}}/2)\to H^{k+i}(X;\mathbb{\mathbb{Z}}/2)$ is trivial for $k>i$ we have the first set of differentials are zero. The second set are trivial because $S{q}^2$ sends $H^3(X;\mathbb{\mathbb{Z}}/2)\to H^5(X)=0$ the identification $S{q}^3=\beta \circ S{q}^2\circ r$ 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 $\lambda \in H^3(X,\mathbb{\mathbb{Z}})$. Then, the spectral sequence reads as

$E_2^{p,q}=H^p(X;K{U}^q(*))\Rightarrow K{U}_{\lambda }^{p+q}(X)$

but with different differentials. For example,

$E_3^{p,q}=E_2^{p,q}=\begin{array}{ccccc}⋮& ⋮& ⋮& ⋮& ⋮\\ 2& H^0(S^3;\mathbb{\mathbb{Z}})& 0& 0& H^3(S^3;\mathbb{\mathbb{Z}})\\ 1& 0& 0& 0& 0\\ 0& H^0(S^3;\mathbb{\mathbb{Z}})& 0& 0& H^3(S^3;\mathbb{\mathbb{Z}})\\ -1& 0& 0& 0& 0\\ -2& H^0(S^3;\mathbb{\mathbb{Z}})& 0& 0& H^3(S^3;\mathbb{\mathbb{Z}})\\ ⋮& ⋮& ⋮& ⋮& ⋮\end{array}$

On the $E_3$-page the differential is

$d_3=S{q}^3+\lambda$

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

$\begin{array}{rl}{d}_5& =\{\lambda ,\lambda ,-\}\\ d_7& =\{\lambda ,\lambda ,\lambda ,-\}\end{array}$

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 $E_{\mathrm{\infty }}=E_4$ 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 $S{q}^3=\beta \circ S{q}^2\circ r$ 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

$K{U}_{\lambda }^k=\{\begin{array}{ll}\mathbb{\mathbb{Z}}& k\text{ is even}\\ \mathbb{\mathbb{Z}}/\lambda & k\text{ is odd}\end{array}$

Rational bordism

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

$\mathbb{\mathbb{Q}}[[{\mathbb{ℂ}\mathbb{\mathbb{P}}}^0],[{\mathbb{ℂ}\mathbb{\mathbb{P}}}^2],[{\mathbb{ℂ}\mathbb{\mathbb{P}}}^4],[{\mathbb{ℂ}\mathbb{\mathbb{P}}}^6],\dots ]$

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

Complex cobordism

Recall that $M{U}^{*}(pt)=\mathbb{\mathbb{Z}}[x_1,x_2,\dots ]$ 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

$E_2^{p,q}=H^p(X;M{U}^q(pt))$

See also

• Quillen–Lichtenbaum conjecture

References

• Davis, James; Kirk, Paul, Lecture Notes in Algebraic Topology, http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf, 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, https://books.google.com/books?id=4hE7AAAAIAAJ&pg=PA197 
• Atiyah, Michael, Twisted K-Theory and cohomology, Bibcode: 2005math.....10674A