Atiyah–Hirzebruch spectral sequence explained

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

E₂

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_Sing

. It can be derived from an exact couple that gives the

E₁

page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with

. In detail, assume

to be the total space of a Serre fibration with fibre

and base space

. The filtration of

by its

-skeletons

B_n

gives rise to a filtration of

. There is a corresponding spectral sequence with

E₂

term

H^p(B;E^q(F))

and converging to the associated graded ring of the filtered ring

	p,q
E
	infty

⇒ E^p+q(X)

This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre

is a point.

Examples

Topological K-theory

For example, the complex topological

-theory of a point is

KU(*)=Z[x,x^-1]

where

is in degree

By definition, the terms on the

E₂

-page of a finite CW-complex

look like

	p,q
E
	2

(X)=H^p(X;KU^q(pt))

Since the

-theory of a point is

K^q(pt)=\begin{cases} Z&ifqiseven\\ 0&otherwise\end{cases}

we can always guarantee that

	p,2k+1
E
	2

(X)=0

This implies that the spectral sequence collapses on

E₂

for many spaces. This can be checked on every

CPⁿ

, algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in

CPⁿ

Cotangent bundle on a circle

For example, consider the cotangent bundle of

S¹

. This is a fiber bundle with fiber

so the

E₂

-page reads as

\begin{array}{c|cc} \vdots&\vdots&\vdots\\ 2&H^0(S^1;Q)&H^1(S^1;Q)\\ 1&0&0\\ 0&H^0(S^1;Q)&H^1(S^1;Q)\\ -1&0&0\\ -2&H^0(S^1;Q)&H^1(S^1;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

d₃

it is the Steenrod square

Sq³

where we take it as the composition

\beta\circSq²\circr

where

is reduction mod

and

\beta

is the Bockstein homomorphism (connecting morphism) from the short exact sequence

0\toZ\toZ\toZ/2\to0

Complete intersection 3-fold

Consider a smooth complete intersection 3-fold

(such as a complete intersection Calabi-Yau 3-fold). If we look at the

E₂

-page of the spectral sequence

\begin{array}{c|ccccc} \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 2&H^0(X;Z)&0&H^2(X;Z)&H^3(X;Z)&H^4(X;Z)&0&H^6(X;Z)\\ 1&0&0&0&0&0&0&0\\ 0&H^0(X;Z)&0&H^2(X;Z)&H^3(X;Z)&H^4(X;Z)&0&H^6(X;Z)\\
-1&0&0&0&0&0&0&0\\ -2&H^0(X;Z)&0&H^2(X;Z)&H^3(X;Z)&H^4(X;Z)&0&H^{6(X;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

	0,2k

	3

\to

	3,2k-2
E
	3

\\ d_3:E

	3,2k

	3

\to

	6,2k-2
E
	3

\end{align}

It turns out that these differentials vanish in both cases, hence

E₂=E_infty

. In the first case, since

Sq^k:H^i(X;Z/2)\toH^k+i(X;Z/2)

is trivial for

k>i

we have the first set of differentials are zero. The second set are trivial because

Sq²

sends

H^3(X;Z/2)\toH^5(X)=0

the identification

Sq³=\beta\circSq²\circr

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_ijg_jkg_ki=λ_ijk

for some cohomology class

λ\inH^3(X,Z)

. Then, the spectral sequence reads as

	p,q
E
	2

=H^p(X;KU^q(*)) ⇒

	p+q
KU
	λ(X)

but with different differentials. For example,

	p,q
E
	3

	p,q
E
	2

=\begin{array}{c|cccc} \vdots&\vdots&\vdots&\vdots&\vdots\\ 2&H^0(S^3;Z)&0&0&H^3(S^3;Z)\\ 1&0&0&0&0\\ 0&H^0(S^3;Z)&0&0&H^3(S^3;Z)\\ -1&0&0&0&0\\ -2&H^0(S^3;Z)&0&0&H^3(S^3;Z)\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \hline&0&1&2&3 \end{array}

On the

E₃

-page the differential is

d₃=Sq³+λ

Higher odd-dimensional differentials

d_2k+1

are given by Massey products for twisted K-theory tensored by

. So

\begin{align} d₅&=\{λ,λ,-\}\\ d₇&=\{λ,λ,λ,-\} \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=E₄

in this case. In particular, this includes all smooth projective varieties.

Twisted K-theory of 3-sphere

The twisted K-theory for

S³

can be readily computed. First of all, since

Sq³=\beta\circSq²\circr

and

H^2(S³⁾=0

, we have that the differential on the

E₃

-page is just cupping with the class given by

. This gives the computation

	k
KU
	λ

=\begin{cases} Z&kiseven\\ Z/λ&kisodd \end{cases}

Rational bordism

Recall that the rational bordism group

	SO
\Omega
	*

⊗ Q

is isomorphic to the ring

Q[[CP^0],[CP^2],[CP^4],[CP^6],\ldots]

generated by the bordism classes of the (complex) even dimensional projective spaces

[CP^2k]

in degree

. This gives a computationally tractable spectral sequence for computing the rational bordism groups.

Complex cobordism

Recall that

MU^*(pt)=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

via the spectral sequence. We have the

E₂

-page given by

	p,q
E
	2

=H^p(X;MU^q(pt))

Atiyah–Hirzebruch spectral sequence explained

Examples

Topological K-theory

Cotangent bundle on a circle

Differentials

Complete intersection 3-fold

Twisted K-theory

Twisted K-theory of 3-sphere

Rational bordism

Complex cobordism

See also