Английская Википедия:Atiyah–Hirzebruch spectral sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by Шаблон:Harvs in the special case of topological K-theory. For a CW complex <math>X</math> and a generalized cohomology theory <math>E^\bullet</math>, it relates the generalized cohomology groups
- <math>E^i(X)</math>
with 'ordinary' cohomology groups <math>H^j</math> with coefficients in the generalized cohomology of a point. More precisely, the <math>E_2</math> term of the spectral sequence is <math>H^p(X;E^q(pt))</math>, and the spectral sequence converges conditionally to <math>E^{p+q}(X)</math>.
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 <math>E=H_{\text{Sing}}</math>. It can be derived from an exact couple that gives the <math>E_1</math> page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with <math>E</math>. In detail, assume <math>X</math> to be the total space of a Serre fibration with fibre <math>F</math> and base space <math>B</math>. The filtration of <math>B</math> by its <math>n</math>-skeletons <math>B_n</math> gives rise to a filtration of <math>X</math>. There is a corresponding spectral sequence with <math>E_2</math> term
- <math>H^p(B; E^q(F))</math>
and converging to the associated graded ring of the filtered ring
- <math>E_\infty^{p,q} = E^{p+q}(X)</math>.
This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre <math>F</math> is a point.
Examples
Topological K-theory
For example, the complex topological <math>K</math>-theory of a point is
- <math>KU(*) = \mathbb{Z}[x,x^{-1}]</math> where <math>x</math> is in degree <math>2</math>
By definition, the terms on the <math>E_2</math>-page of a finite CW-complex <math>X</math> look like
- <math>E_2^{p,q}(X) = H^p(X;KU^q(pt))</math>
Since the <math>K</math>-theory of a point is
- <math>
K^q(pt) = \begin{cases} \mathbb{Z} & \text{if q is even} \\ 0 & \text{otherwise} \end{cases} </math> we can always guarantee that
- <math>E_2^{p,2k+1}(X) = 0</math>
This implies that the spectral sequence collapses on <math>E_2</math> for many spaces. This can be checked on every <math>\mathbb{CP}^n</math>, algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in <math>\mathbb{CP}^n</math>.
Cotangent bundle on a circle
For example, consider the cotangent bundle of <math>S^1</math>. This is a fiber bundle with fiber <math>\mathbb{R}</math> so the <math>E_2</math>-page reads as
- <math>
\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} </math>
Differentials
The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For <math>d_3</math> it is the Steenrod square <math>Sq^3</math> where we take it as the composition
- <math> \beta \circ Sq^2 \circ r</math>
where <math>r</math> is reduction mod <math>2</math> and <math>\beta</math> is the Bockstein homomorphism (connecting morphism) from the short exact sequence
- <math>0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2 \to 0</math>
Complete intersection 3-fold
Consider a smooth complete intersection 3-fold <math>X</math> (such as a complete intersection Calabi-Yau 3-fold). If we look at the <math>E_2</math>-page of the spectral sequence
- <math>
\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} </math> we can see immediately that the only potentially non-trivial differentials are
- <math>
\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} </math> It turns out that these differentials vanish in both cases, hence <math>E_2 = E_\infty</math>. In the first case, since <math>Sq^k:H^i(X;\mathbb{Z}/2) \to H^{k+i}(X;\mathbb{Z}/2)</math> is trivial for <math>k > i</math> we have the first set of differentials are zero. The second set are trivial because <math>Sq^2</math> sends <math>H^3(X;\mathbb{Z}/2) \to H^5(X) = 0</math> the identification <math>Sq^3 = \beta \circ Sq^2 \circ r</math> 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 <math>(U_{ij},g_{ij})</math> where
- <math> g_{ij}g_{jk}g_{ki} = \lambda_{ijk} </math>
for some cohomology class <math>\lambda \in H^3(X,\mathbb{Z})</math>. Then, the spectral sequence reads as
- <math> E_2^{p,q} = H^p(X;KU^q(*)) \Rightarrow KU^{p+q}_\lambda(X)</math>
but with different differentials. For example,
- <math>
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} </math> On the <math>E_3</math>-page the differential is
- <math> d_3 = Sq^3 + \lambda </math>
Higher odd-dimensional differentials <math>d_{2k+1}</math> are given by Massey products for twisted K-theory tensored by <math>\mathbb{R}</math>. So
- <math>
\begin{align} d_5 &= \{ \lambda, \lambda, - \} \\ d_7 &= \{ \lambda, \lambda, \lambda, - \} \end{align} </math> 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 <math>E_\infty = E_4</math> in this case. In particular, this includes all smooth projective varieties.
Twisted K-theory of 3-sphere
The twisted K-theory for <math>S^3</math> can be readily computed. First of all, since <math>Sq^3 = \beta \circ Sq^2 \circ r</math> and <math>H^2(S^3) = 0</math>, we have that the differential on the <math>E_3</math>-page is just cupping with the class given by <math>\lambda</math>. This gives the computation
- <math> KU_\lambda^k = \begin{cases}
\mathbb{Z} & k \text{ is even} \\ \mathbb{Z}/\lambda & k \text{ is odd} \end{cases} </math>
Rational bordism
Recall that the rational bordism group <math>\Omega_*^{\text{SO}}\otimes \mathbb{Q}</math> is isomorphic to the ring
- <math> \mathbb{Q}[[\mathbb{CP}^0], [\mathbb{CP}^2], [\mathbb{CP}^4],[\mathbb{CP}^6],\ldots]</math>
generated by the bordism classes of the (complex) even dimensional projective spaces <math>[\mathbb{CP}^{2k}]</math> in degree <math>4k</math>. This gives a computationally tractable spectral sequence for computing the rational bordism groups.
Complex cobordism
Recall that <math>MU^*(pt) = \mathbb{Z}[x_1,x_2,\ldots]</math> where <math>x_i \in \pi_{2i}(MU)</math>. Then, we can use this to compute the complex cobordism of a space <math>X</math> via the spectral sequence. We have the <math>E_2</math>-page given by
- <math>E_2^{p,q} = H^p(X;MU^q(pt))</math>
See also
References
