nLab Atiyah–Hirzebruch spectral sequence

Idea

The Atiyah-Hirzebruch spectral sequence (AHSS) is a type of spectral sequence that generalizes the Serre spectral sequence from ordinary cohomology $H^\bullet$ to any generalized (Eilenberg-Steenrod) cohomology theory $E^\bullet$.

For any (finite) homotopy fiber sequence

$$\array{ F &\longrightarrow& P \\ && \downarrow \\ && X }$$

then the corresponding $E$-Atiyah-Hirzebruch spectral sequence has on its second page the ordinary cohomology of $X$ with coefficients in the $E$-cohomology groups of the fiber and converges to the proper $E$-cohomology of the total space:

$$E_2^{p,q} = H^p(X,E^q(F)) \Rightarrow E^{\bullet}(P) \,.$$

This is of interest already for $F \simeq \ast$, as then it expresses generalized cohomology in terms of ordinary cohomology with coefficients in the base cohomology ring.

Remark

(note on terminology)

Often the terminology “Atiyah-Hirzebruch spectral sequence” is taken to refer to only this case with $F = \ast$, while the general case is then referred to as “Serre spectral sequence for generalized cohomology” or similar. In (Atiyah-Hirzebruch 61,p. 17) the case $F = \ast$ is labeled “Theorem”, while the general case, stated right after the theorem, is labeled “2.2 Remark”.

The proof of the theorem that is given is very short, it just says that since topological K-theory satisfies the exactness axiom of a generalized cohomology theory, it is immediate that the conditions for a spectral sequence stated as Axioms (SP.1)-(SP.5) in (Cartan-Eilenberg 56, section XV.7) are met. Indeed Example 2 in (Cartan-Eilenberg 56, section XV.7) observes that the spectral sequence in question exists for $E$ “some fixed cohomology theory” because “Axioms (SP.1)-(SP.4) are consequences of usual properties of cohomology groups”.

In view of this, the contribution of (Atiyah-Hirzebruch 61) would not be so much the observation of what is now called the AHSS, rather than the proof that K-theory satisfies the axioms of a generalized cohomology theory. Indeed, according to (Adams 74, p. 127-128, 215), the AHSS was earlier observed by George Whitehead and “then became a folk-theorem” which was “eventually published by Atiyah and Hirzebruch”.

Maybe it should be called the “Cartan-Eilenberg-Whitehead spectral sequence”.

There is a generalization to equivariant cohomology theory (Davis-Lueck 98, theorem 4.7).

For genuine G-equivariance, with RO(G)-grading for representation spheres $S^V$, then for $P \to X$ an $F$-fibration of topological G-spaces and for $A$ any $G$-Mackey functor, the equivariant Serre spectral sequence looks like (Kronholm 10, theorem 3.1):

$$E_2^{p,q} = H^p(X, H^{V+q}(F,A)) \,\Rightarrow\, H^{V+p+q}(P,A) \,,$$

where on the left in the $E_2$-page we have ordinary cohomology with coefficients in the genuine equivariant cohomology groups of the fiber.

Construction

Statement

Proposition

Let $A^\bullet$ be a an additive unreduced generalized cohomology functor (def.). Let $B$ be a CW-complex and let $X \stackrel{\pi}{\to} B$ be a Serre fibration, such that all its fibers are weakly contractible or such that $B$ is simply connected. In either case all fibers are identified with a typical fiber $F$ up to weak homotopy equivalence by connectedness (this example), and well-defined up to unique iso in the homotopy category by simply-connectedness:

$$\array{ F &\longrightarrow& X \\ && \Big\downarrow\mathrlap{{}^{\in Fib_{cl}}} \\ && B } \,.$$

If at least one of the following two conditions is met

• $B$ is finite-dimensional as a CW-complex;

• $A^\bullet(F)$ is bounded below in degree and the sequences $\cdots \to A^p(X_{n+1}) \to A^p(X_n) \to \cdots$ satisfy the Mittag-Leffler condition (def.) for all $p$;

then there is a cohomology spectral sequence, (def.), whose $E_2$-page is the ordinary cohomology $H^\bullet(B,A^\bullet(F))$ of $B$ with coefficients in the $A$-cohomology groups $A^\bullet(F)$ of the fiber, and which converges to the $A$-cohomology groups of the total space

$$E_2^{p,q} = H^p(B, A^q(F)) \; \Rightarrow \; A^\bullet(X)$$

with respect to the filtering given by

$$F^p A^\bullet(X) \coloneqq ker\left( A^\bullet(X) \to A^\bullet(X_{p-1}) \right) \,,$$

where $X_{p} \coloneqq \pi^{-1}(B_{p})$ is the fiber over the $p$th stage of the CW-complex $B = \underset{\longleftarrow}{\lim}_n B_n$.

Generally, without assumptions on the connectivity of $B$, there is a spectral sequence of this form with ordinary cohomology with coefficients in $A^\bullet(F)$ replaced by ordinary cohomology with local coefficients $(b \mapsto A^\bullet(F_b))$.

Construction by filtering the base space

The following proof is the standard and original argument due to (Atiyah-Hirzebruch 61, p. 17).

Proof

The exactness axiom for $A$ gives an exact couple, (def.), of the form

$$\array{ \underset{s,t}{\prod} A^{s+t}(X_{s}) && \stackrel{}{\longrightarrow} && \underset{s,t}{\prod} A^{s+t}(X_{s}) \\ & \nwarrow && \swarrow \\ && \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1}) } \;\;\;\;\;\;\; \left( \array{ A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1}) \\ \uparrow && \downarrow_{\mathrlap{\delta}} \\ A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1}) } \right) \,,$$

where we take $X_{\gg 1} = X$ and $X_{\lt 0} = \emptyset$.

In order to determine the $E_2$-page, we analyze the $E_1$-page: By definition

$$E_1^{s,t} = A^{s+t}(X_s, X_{s-1})$$

Let $C(s)$ be the set of $s$-dimensional cells of $B$, and notice that for $\sigma \in C(s)$ then

$$(\pi^{-1}(\sigma), \pi^{-1}(\partial \sigma)) \simeq (D^n, S^{n-1}) \times F_\sigma \,,$$

where $F_\sigma$ is weakly homotopy equivalent to $F$ (exmpl.).

This implies that

$$\begin{aligned} E_1^{s,t} & \coloneqq A^{s+t}(X_s, X_{s-1}) \\ & \simeq \tilde A^{s+t}(X_s/X_{s-1}) \\ & \simeq \tilde A^{s+t}(\underset{\sigma \in C(n)}{\vee} S^s \wedge F_+) \\ & \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} \tilde A^{s+t}(S^s \wedge F_+) \\ & \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} \tilde A^t(F_+) \\ & \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} A^t(F) \\ & \simeq C^s_{cell}(B,A^t(F)) \end{aligned} \,,$$

where we used the relation to reduced cohomology $\tilde A$, (prop.) together with (lemma), then the wedge axiom and the suspension isomorphism of the latter.

The last group $C^s_{cell}(B,A^t(F))$ appearing in this sequence of isomorphisms is that of cellular cochains (def.) of degree $s$ on $B$ with coefficients in the group $A^t(F)$.

Since cellular cohomology of a CW-complex agrees with its singular cohomology (thm.), hence with its ordinary cohomology, to conclude that the $E_2$-page is as claimed, it is now sufficient to show that the differential $d_1$ coincides with the differential in the cellular cochain complex (def.).

We discuss this now for $\pi = id$, hence $X = B$ and $F = \ast$. (The general case works the same, just with various factors of $F$ replacing the point.)

Consider the following diagram, which commutes due to the naturality of the connecting homomorphism $\delta$ of $A^\bullet$:

$$\array{ \partial^\ast \colon & C^{s-1}_{cell}(X,A^t(\ast)) & =& \underset{i \in I_{s-1}}{\prod} A^t(\ast) && \longrightarrow && \underset{i \in I_s}{\prod} A^t(\ast) & = & C_{cell}^{s}(X,A^t(\ast)) \\ && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}} \\ && & \underset{i \in I_{s-1}}{\prod} \tilde A^{s+t-1}(S^{s-1}) && && \underset{i \in I_s}{\prod} \tilde A^{s+t}(S^{s}) \\ && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}} \\ && d_1 \colon & A^{s+t-1}(X_{s-1}, X_{s-2}) &\overset{}{\longrightarrow}& A^{s+t-1}(X_{s-1}) &\overset{\delta}{\longrightarrow}& A^{s+t}(X_s, X_{s-1}) \\ && & \downarrow && \downarrow && \downarrow \\ && & A^{s+t-1}(S^{s-1}, \emptyset) &\overset{}{\longrightarrow}& A^{s+t-1}(S^{s-1}) &\overset{\delta}{\longrightarrow}& A^{s+t}(D^s , S^{s-1}) } \,.$$

Here the bottom vertical morphisms are those induced from any chosen cell inclusion $(D^s , S^{s-1}) \hookrightarrow (X_s, X_{s-1})$.

The differential $d_1$ in the spectral sequence is the middle horizontal composite. From this the vertical isomorphisms give the top horizontal map. But the bottom horizontal map identifies this top horizontal morphism componentwise with the restriction to the boundary of cells. Hence the top horizontal morphism is indeed the coboundary operator $\partial^\ast$ for the cellular cohomology of $X$ with coefficients in $A^\bullet(\ast)$ (def.). This cellular cohomology coincides with singular cohomology of the CW-complex $X$ (thm.), hence computes the ordinary cohomology of $X$.

Now to see the convergence. If $B$ is finite dimensional then the convergence condition as stated in this prop. is met. Alternatively, if $A^\bullet(F)$ is bounded below in degree, then by the above analysis the $E_1$-page has a horizontal line below which it vanishes. Accordingly the same is then true for all higher pages, by each of them being the cohomology of the previous page. Since the differentials go right and down, eventually they pass beneath this vanishing line and become 0. This is again the condition needed in the proof of this prop. to obtain convergence.

By that proposition the convergence is to the inverse limit

$$\underset{\longleftarrow}{\lim} \left( \cdots \stackrel{}{\to} A^\bullet(X_{s+1}) \longrightarrow A^\bullet(X_{s}) \to \cdots \right) \,.$$

If $X$ is finite dimensional or more generally if the sequences that this limit is over satisfy the Mittag-Leffler condition (def.), then this limit is $A^\bullet(X)$, by this prop..

Construction by filtering the coefficient spectrum

One also gets the Atiyah-Hirzebruch spectral sequence, up to isomorphism, by instead using the filtering given by the Postnikov tower of an Omega-spectrum representing the given generalized cohomology theory.

This is due to (Maunder 63, theorem 3.3)

This alternative construction should be the one that is discussed in (Shulman 13) from the perspective of homotopy type theory.

tower diagram/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

Properties

Multiplicative structure

for $E$ a ring spectrum, then the AHSS is multiplicative…

(MO discussion)

Kronecker pairing

Proposition

For $E$ a ring spectrum and $X$ a CW complex of finite dimension, then the Kronecker pairing $\langle -,-\rangle \colon E^\bullet(X)\otimes E_\bullet(X)\to \pi_\bullet(E)$ passes to a page-wise pairing of the corresponding Atiyah-Hirzebruch spectral sequences for $E$-cohomology/homology

$$\langle-,-\rangle_r \;\colon\; \mathcal{E}_r^{n,-s} \otimes \mathcal{E}^r_{n,t} \longrightarrow \pi_{s+t}(E)$$

such that

1. on the $\mathcal{E}_2$-page this restricts to the Kronecker pairing for ordinary cohomology/ordinary homology with coefficients in $\pi_\bullet(E)$;

2. the differentials act as derivations

   $$\langle d_r(-),-\rangle = \langle -, d^r(-)\rangle \,,$$

3. The pairing on the $\mathcal{E}_\infty$-page is compatible with the Kronecker pairing.

(Kochman 96, prop. 4.2.10)

Examples and Applications

To complex oriented cohomology theory

…(Adams 74)….

For topological K-theory

For $E =$ KU, hence for topological K-theory, the differential $d_3$ of the Atiyah-Hirzebruch spectral sequence with $F = \ast$ is given by a Steenrod square operation

$$d_3 = Sq^3$$

(Atiyah-Hirzebruch 61).

For twisted K-theory this picks up in addition the cup product with the 3-class $H$ of the twist:

$$d_3 = Sq^3 + H \cup(-)$$

(Rosenberg 82, Atiyah-Segal 05 (4.1)). The higher differentials $d_5$ and $d_7$ here are given by higher Massey products with the twisting class (Atiyah-Segal 05 sections 5-7).

D-brane charges in string theory

In string theory D-brane charges are classes in $E = KU$-cohomology, i.e. in K-theory. The second page of of the corresponding Atiyah-Hirzebruch spectral sequence (see above) for $F = \ast$ hence expresses ordinary cohomology in all even or all odd degrees, and being in the kernel of all the differentials is hence the constraint on such ordinary cohomology data to lift to genuine K-theory classes, hence to genuine D-brane charges. In this way the Atiyah-Hirzebruch spectral sequences is used in (Maldacena-Moore-Seiberg 01, Evslin-Sati 06)

References

General

The statement of the existence of the spectral sequence first appears in print (with $E =$ topological K-theory) in

• M. F. Atiyah, F. Hirzebruch, Vector bundles and homogeneous spaces, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I. (web, doi:10.1142/9789814401319_0008, MR 0139181)

but the proof given consists essentially in pointing to section XV.7 (“a more general setting in which the theory of spectral sequences may be developed”) of

• Henri Cartan, Samuel Eilenberg, Homological algebra, Princeton Univ. Press (1956)

This is by filtering over the stages of the base space CW-complex. That one gets an isomorphic spectral sequence by instead filtering over the Postnikov tower of any Omega-spectrum representing the given generalized cohomology theory is due to

• C. R. F. Maunder, The spectral sequence of an extraordinary cohomology theory, Mathematical Proceedings of the Cambridge Philosophical Society 59 3 (1963) 567- 574 $[$doi:10.1017/S0305004100037245$]$

Early lecture notes:

• Frank Adams, part III section 7 of Stable homotopy and generalised homology, 1974

where the idea is attributed to George Whitehead:

the Atiyah-Hirzebruch spectral sequence, which was really invented by G. W. Whitehead but not published by him (Adams 74, p. 127-128)

These spectral sequences were probably first invented by G. W. Whitehead, but he got them just after he wrote the paper $[$Whitehead 56 ‘Homotopy groups of joins and unions’$]$ in which they ought to have appeared. They then became a folk-theorem and were eventually published by Atiyah and Hirzebruch (Adams 74, p. 215)

A more detailed account of the proof is in

• Stanley Kochman, theorem 4.2.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

Further discussion of the case of twisted K-theory:

• Jonathan Rosenberg, Homological Invariants of Extensions of $C^\ast$-algebras, Proc. Symp. Pure Math 38 (1982) 35.

• Michael Atiyah, Graeme Segal, Twisted K-theory and cohomology, Nankai Tracts Math. 11, World Sci. Publ., Hackensack, NJ, pp. 5–43, (arXiv:math/0510674)

• Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Section 21 of: Basic Bundle Theory and K-Cohomology Invariants, Springer Lecture Notes in Physics 726, 2008, (pdf, doi:10.1007/978-3-540-74956-1)

An analogous spectral sequence for higher twisted de Rham cohomology is discussed in Li, Liu & Wang 2009.

A discussion of the $E$-AHSS as the spectral sequence of a tower induced by forming mapping spectra $[X,-]$ into the Postnikov tower is due to

• Mike Shulman, spectral sequences (2013)

Equivariant version

Discussion of the AHSS in Bredon equivariant stable homotopy theory/equivariant cohomology includes

• James Davis, Wolfgang Lueck, Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory, K-Theory 15: 201–252, 1998 (pdf, K-Theory archive)

Discussion in genuine equivariant cohomology, i.e. including RO(G)-grading, is in

• William Kronholm, The $RO(G)$-graded Serre spectral sequence, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. (pdf, Euclid)

Examples

Application for the case of K-theory to D-brane charges in string theory is discussed in

• Juan Maldacena, Gregory Moore, Nathan Seiberg, D-Brane Instantons and K-Theory Charges, JHEP 0111:062,2001 (arXiv:hep-th/0108100)

• Jarah Evslin, Hisham Sati, Can D-Branes Wrap Nonrepresentable Cycles?, JHEP0610:050,2006 (arXiv:hep-th/0607045)

and detailed review of this is in

• Fabio Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)

Discussion for the $\mathbb{Z}/2$-equivariant KR cohomology theory (relevant for D-branes in orientifolds) includes

• Daniel Dugger, An Atiyah-Hirzebruch spectral sequence for KR-theory (arXiv:math/0304099

Discussion for the case of Morava K-theory and Morava E-theory with comments on application to charges of M-branes is in

• Hisham Sati, Craig Westerland, Twisted Morava K-theory and E-theory (arXiv:1109.3867)

Application to motivic cobordism cohomology theory is discussed in

• Nobuaki Yagita, Applications of the Atiyah-Hirzebruch spectral sequences for motivic cobordisms (pdf)

Application to the K-theory classification of topological phases of matter:

• Ken Shiozaki, Masatoshi Sato, Kiyonori Gomi, Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups, Phys. Rev. B 106 (2022) 165103 [arXiv:1802.06694, doi:10.1103/PhysRevB.106.165103]