# Adams spectral sequence

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Spectral sequence

In [mathematics](/source/Mathematics), the **Adams spectral sequence** is a [spectral sequence](/source/Spectral_sequence) introduced by [J. Frank Adams](/source/Frank_Adams) ([1958](#CITEREFAdams1958)) which computes the [stable homotopy groups](/source/Stable_homotopy_theory) of [topological spaces](/source/Topological_spaces). Like all spectral sequences, it is a computational tool; it relates [homology](/source/Homology_(mathematics)) theory to what is now called [stable homotopy theory](/source/Stable_homotopy_theory). It is a reformulation using [homological algebra](/source/Homological_algebra), and an extension, of a technique called 'killing homotopy groups' applied by the French school of [Henri Cartan](/source/Henri_Cartan) and [Jean-Pierre Serre](/source/Jean-Pierre_Serre).

## Motivation

For everything below, once and for all, we fix a prime *p*. All spaces are assumed to be [CW complexes](/source/CW_complex). The [ordinary](/source/Singular_cohomology) [cohomology groups](/source/Cohomology_group) H ∗ ( X ) {\displaystyle H^{*}(X)} are understood to mean H ∗ ( X ; Z / p Z ) {\displaystyle H^{*}(X;\mathbb {Z} /p\mathbb {Z} )} .

The primary goal of [algebraic topology](/source/Algebraic_topology) is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces *X* and *Y*. This is extraordinarily ambitious: in particular, when *X* is S n {\displaystyle S^{n}} , these maps form the *n*th [homotopy group](/source/Homotopy_group) of *Y*. A more reasonable (but still very difficult!) goal is to understand the set [ X , Y ] {\displaystyle [X,Y]} of maps (up to homotopy) that remain after we apply the [suspension functor](/source/Suspension_(topology)) a large number of times. We call this the collection of stable maps from *X* to *Y*. (This is the starting point of [stable homotopy theory](/source/Stable_homotopy_theory); more modern treatments of this topic begin with the concept of a [spectrum](/source/Spectrum_(homotopy_theory)). Adams' original work did not use spectra, and we avoid further mention of them in this section to keep the content here as elementary as possible.)

The set [ X , Y ] {\displaystyle [X,Y]} turns out to be an abelian group, and if *X* and *Y* are reasonable spaces this group is finitely generated. To figure out what this group is, we first isolate a prime *p*. In an attempt to compute the *p*-torsion of [ X , Y ] {\displaystyle [X,Y]} , we look at cohomology: send [ X , Y ] {\displaystyle [X,Y]} to Hom(*H**(*Y*), *H**(*X*)). This is a good idea because cohomology groups are usually tractable to compute.

The key idea is that H ∗ ( X ) {\displaystyle H^{*}(X)} is more than just a graded [abelian group](/source/Abelian_group), and more still than a graded [ring](/source/Ring_(mathematics)) (via the [cup product](/source/Cup_product)). The representability of the cohomology functor makes *H**(*X*) a [module](/source/Module_(mathematics)) over the algebra of its stable [cohomology operations](/source/Cohomology_operation), the [Steenrod algebra](/source/Steenrod_algebra) *A*. Thinking about *H**(*X*) as an *A*-module forgets some cup product structure, but the gain is enormous: Hom(*H**(*Y*), *H**(*X*)) can now be taken to be *A*-linear! A priori, the *A*-module sees no more of [*X*, *Y*] than it did when we considered it to be a map of vector spaces over F*p*. But we can now consider the derived functors of Hom in the category of *A*-modules, [Ext](/source/Ext_functor)*A**r*(*H**(*Y*), *H**(*X*)). These acquire a second grading from the grading on *H**(*Y*), and so we obtain a two-dimensional "page" of algebraic data. The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step.

The point of all this is that *A* is so large that the above sheet of cohomological data contains all the information we need to recover the *p*-primary part of [*X*, *Y*], which is homotopy data. This is a major accomplishment because cohomology was designed to be computable, while homotopy was designed to be powerful. This is the content of the Adams spectral sequence.

## Classical formulation

### Formulation for computing homotopy groups of spectra

The classical Adams spectral sequence can be stated for any [connective spectrum](/source/Connective_spectrum) X {\displaystyle X} of [finite type](https://en.wikipedia.org/w/index.php?title=Spectrum_of_finite_type&action=edit&redlink=1), meaning π i ( X ) = 0 {\displaystyle \pi _{i}(X)=0} for i < 0 {\displaystyle i<0} and π i ( X ) {\displaystyle \pi _{i}(X)} is a finitely generated Abelian group in each degree. Then, there is a spectral sequence E ∗ ∗ , ∗ ( X ) {\displaystyle E_{*}^{*,*}(X)} [1]: 41 such that

1. E 2 s , t = Ext A p s , t ( H ∗ ( X ) , Z / p ) {\displaystyle E_{2}^{s,t}={\text{Ext}}_{A_{p}}^{s,t}(H^{*}(X),\mathbb {Z} /p)} for A p {\displaystyle A_{p}} the mod p {\displaystyle p} Steenrod algebra

1. For X {\displaystyle X} of finite type, E ∞ ∗ , ∗ {\displaystyle E_{\infty }^{*,*}} is a bigraded group associated with a filtration of π ∗ ( X ) ⊗ Z p {\displaystyle \pi _{*}(X)\otimes \mathbb {Z} _{p}} (the [p-adic integers](/source/P-adic_integers))

Note that this implies for X = S {\displaystyle X=\mathbb {S} } , this computes the p {\displaystyle p} -torsion of the homotopy groups of the [sphere spectrum](/source/Sphere_spectrum), i.e. the stable homotopy groups of the spheres. Also, because for any CW-complex Y {\displaystyle Y} we can consider the suspension spectrum Σ ∞ Y {\displaystyle \Sigma ^{\infty }Y} , this gives the statement of the previous formulation as well.

This statement generalizes a little bit further by replacing the A p {\displaystyle {\mathcal {A}}_{p}} -module Z / p {\displaystyle \mathbb {Z} /p} with the cohomology groups H ∗ ( Y ) {\displaystyle H^{*}(Y)} for some connective spectrum Y {\displaystyle Y} (or topological space Y {\displaystyle Y} ). This is because the construction of the spectral sequence uses a "free" resolution of H ∗ ( X ) {\displaystyle H^{*}(X)} as an A p {\displaystyle {\mathcal {A}}_{p}} -module, hence we can compute the Ext groups with H ∗ ( Y ) {\displaystyle H^{*}(Y)} as the second entry. We therefore get a spectral sequence with E 2 {\displaystyle E_{2}} -page given by

E 2 t , s = Ext A p s , t ( H ∗ ( X ) , H ∗ ( Y ) ) {\displaystyle E_{2}^{t,s}={\text{Ext}}_{{\mathcal {A}}_{p}}^{s,t}(H^{*}(X),H^{*}(Y))}

which has the convergence property of being isomorphic to the graded pieces of a filtration of the p {\displaystyle p} -torsion of the stable homotopy group of homotopy classes of maps between X {\displaystyle X} and Y {\displaystyle Y} , that is

E 2 s , t ⇒ π t − s S ( [ X , Y ] ) ⊗ Z p {\displaystyle E_{2}^{s,t}\Rightarrow \pi _{t-s}^{\mathbb {S} }([X,Y])\otimes \mathbb {Z} _{p}}

### Spectral sequence for the stable homotopy groups of spheres

For example, if we let both spectra be the sphere spectrum, so X = Y = S {\displaystyle X=Y=\mathbb {S} } , then the Adams spectral sequence has the convergence property

E 2 t , s = Ext A p s , t ( H ∗ ( S ) , H ∗ ( S ) ) ⇒ π t − s ( S ) ⊗ Z p {\displaystyle E_{2}^{t,s}={\text{Ext}}_{{\mathcal {A}}_{p}}^{s,t}(H^{*}(\mathbb {S} ),H^{*}(\mathbb {S} ))\Rightarrow \pi _{t-s}(\mathbb {S} )\otimes \mathbb {Z} _{p}}

giving a technical tool for approaching a computation of the stable homotopy groups of spheres. It turns out that many of the first terms can be computed explicitly from purely algebraic information[2]pp 23–25. Also note that we can rewrite H ∗ ( S ) = Z / p {\displaystyle H^{*}(\mathbb {S} )=\mathbb {Z} /p} , so the E 2 {\displaystyle E_{2}} -page is

E 2 t , s = Ext A p s , t ⁡ ( Z / p , Z / p ) ⇒ π t − s ( S ) ⊗ Z p {\displaystyle E_{2}^{t,s}=\operatorname {Ext} _{{\mathcal {A}}_{p}}^{s,t}(\mathbb {Z} /p,\mathbb {Z} /p)\Rightarrow \pi _{t-s}(\mathbb {S} )\otimes \mathbb {Z} _{p}}

We include this calculation information below for p = 2 {\displaystyle p=2} .

### Ext terms from the resolution

Given the [Adams resolution](/source/Adams_resolution)

⋯ → H ∗ ( F 2 ) → H ∗ ( F 1 ) → H ∗ ( F 0 ) → H ∗ ( X ) {\displaystyle \cdots \to H^{*}(F_{2})\to H^{*}(F_{1})\to H^{*}(F_{0})\to H^{*}(X)}

we have the E 1 {\displaystyle E_{1}} -terms as

E 1 s , t = Hom A p t ⁡ ( H ∗ ( F s ) , H ∗ ( Y ) ) {\displaystyle E_{1}^{s,t}=\operatorname {Hom} _{{\mathcal {A}}_{p}}^{t}(H^{*}(F_{s}),H^{*}(Y))}

for the graded Hom-groups. Then the E 1 {\displaystyle E_{1}} -page can be written as

E 1 = 3 ⋮ ⋮ ⋮ 2 Hom 2 ( H ∗ ( F 0 ) , H ∗ ( Y ) ) Hom 2 ( H ∗ ( F 1 ) , H ∗ ( Y ) ) Hom 2 ( H ∗ ( F 2 ) , H ∗ ( Y ) ) ⋯ 1 Hom 1 ( H ∗ ( F 0 ) , H ∗ ( Y ) ) Hom 1 ( H ∗ ( F 1 ) , H ∗ ( Y ) ) Hom 1 ( H ∗ ( F 2 ) , H ∗ ( Y ) ) ⋯ 0 Hom 0 ( H ∗ ( F 0 ) , H ∗ ( Y ) ) Hom 0 ( H ∗ ( F 1 ) , H ∗ ( Y ) ) Hom 0 ( H ∗ ( F 2 ) , H ∗ ( Y ) ) ⋯ 0 1 2 {\displaystyle E_{1}={\begin{array}{c|ccc}3&\vdots &\vdots &\vdots \\2&{\text{Hom}}^{2}(H^{*}(F_{0}),H^{*}(Y))&{\text{Hom}}^{2}(H^{*}(F_{1}),H^{*}(Y))&{\text{Hom}}^{2}(H^{*}(F_{2}),H^{*}(Y))&\cdots \\1&{\text{Hom}}^{1}(H^{*}(F_{0}),H^{*}(Y))&{\text{Hom}}^{1}(H^{*}(F_{1}),H^{*}(Y))&{\text{Hom}}^{1}(H^{*}(F_{2}),H^{*}(Y))&\cdots \\0&{\text{Hom}}^{0}(H^{*}(F_{0}),H^{*}(Y))&{\text{Hom}}^{0}(H^{*}(F_{1}),H^{*}(Y))&{\text{Hom}}^{0}(H^{*}(F_{2}),H^{*}(Y))&\cdots \\\hline &0&1&2\end{array}}}

so the degree of s {\displaystyle s} can be thought of how "deep" in the Adams resolution we go before we can find the generators.

## Calculations

The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.

### Grading of the Differential

The r {\displaystyle r} th Adams differential always goes to the left 1, and up r {\displaystyle r} . That is,

d r : E r s , t → E r s − 1 , t + r {\displaystyle d_{r}\colon E_{r}^{s,t}\to E_{r}^{s-1,t+r}} .

### Examples with Eilenberg–Maclane spectra

Some of the simplest calculations are with [Eilenberg–Maclane spectra](/source/Eilenberg%E2%80%93Maclane_spectrum) such as X = H Z {\displaystyle X=H\mathbb {Z} } and X = H Z / ( p k ) {\displaystyle X=H\mathbb {Z} /(p^{k})} .[1]: 48 For the first case, we have the E 1 {\displaystyle E_{1}} page

E 1 s , t = { Z / p if t = s 0 otherwise {\displaystyle E_{1}^{s,t}={\begin{cases}\mathbb {Z} /p&{\text{ if }}t=s\\0&{\text{ otherwise }}\end{cases}}}

giving a collapsed spectral sequence, hence E 1 = E ∞ {\displaystyle E_{1}=E_{\infty }} . This can be rewritten as

Ext A p s , t ( H ∗ ( H Z ) , Z / p ) = { Z / p if t = s 0 if t ≠ s {\displaystyle {\text{Ext}}_{{\mathcal {A}}_{p}}^{s,t}(H^{*}(H\mathbb {Z} ),\mathbb {Z} /p)={\begin{cases}\mathbb {Z} /p&{\text{ if }}t=s\\0&{\text{ if }}t\neq s\end{cases}}}

giving the E 2 {\displaystyle E_{2}} -page. For the other case, note there is a cofiber sequence

H Z → ⋅ p k H Z → H Z / p k → Σ H Z {\displaystyle H\mathbb {Z} \xrightarrow {\cdot p^{k}} H\mathbb {Z} \to H\mathbb {Z} /p^{k}\to \Sigma H\mathbb {Z} }

which ends up giving a splitting in cohomology, so H ∗ ( H Z / p k ) = H ∗ ( H Z ) ⊕ H ∗ ( Σ H Z ) {\displaystyle H^{*}(H\mathbb {Z} /p^{k})=H^{*}(H\mathbb {Z} )\oplus H^{*}(\Sigma H\mathbb {Z} )} as A p {\displaystyle {\mathcal {A}}_{p}} -modules. Then, the E 2 {\displaystyle E_{2}} -page of H ∗ ( H Z / p ) {\displaystyle H^{*}(H\mathbb {Z} /p)} can be read as

E 2 s , t = { Z / p if t − s = 0 , 1 0 otherwise {\displaystyle E_{2}^{s,t}={\begin{cases}\mathbb {Z} /p&{\text{if }}t-s=0,1\\0&{\text{otherwise }}\end{cases}}}

The expected E ∞ {\displaystyle E_{\infty }} -page is

E ∞ s , t = { Z / p k if t = s 0 otherwise {\displaystyle E_{\infty }^{s,t}={\begin{cases}\mathbb {Z} /p^{k}&{\text{ if }}t=s\\0&{\text{ otherwise }}\end{cases}}} .

The only way for this spectral sequence to converge to this page is if is there are non-trivial differentials supported on every element with Adams grading ( s , s + 1 ) {\displaystyle (s,s+1)} .

### Other applications

Adams' original use for his spectral sequence was the first proof of the [Hopf invariant](/source/Hopf_invariant) 1 problem: R n {\displaystyle \mathbb {R} ^{n}} admits a [division algebra](/source/Division_algebra) structure only for *n* = 1, 2, 4, or 8. He subsequently found a much shorter proof using cohomology operations in [K-theory](/source/K-theory).

The [Thom isomorphism theorem](/source/Thom_space) relates [differential topology](/source/Differential_topology) to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960, [John Milnor](/source/John_Milnor) and [Sergei Novikov](/source/Sergei_Novikov_(mathematician)) used the Adams spectral sequence to compute the coefficient ring of [complex cobordism](/source/Complex_cobordism). Further, Milnor and [C. T. C. Wall](/source/C._T._C._Wall) used the spectral sequence to prove Thom's conjecture on the structure of the oriented [cobordism](/source/Cobordism) ring: two oriented manifolds are cobordant [if and only if](/source/If_and_only_if) their [Pontryagin](/source/Pontryagin_class#Pontryagin_numbers) and [Stiefel–Whitney numbers](/source/Stiefel%E2%80%93Whitney_class#Stiefel–Whitney_numbers) agree.

## Stable homotopy groups of spheres

Visual diagram demonstrating the

          E

            2

    {\displaystyle E_{2}}

 page of the Adams spectral sequence computing the stable homotopy groups of spheres. The dots represent elements left over from the

          E

            1

    {\displaystyle E_{1}}

 page and the diagonal lines moving up and to the left represent various differentials in the spectral sequence. The differential

          d

            r

    {\displaystyle d_{r}}

 moves one unit to the left and

        r

    {\displaystyle r}

 units upward. The vertical lines are used as a book-keeping tool for determining the structure of the torsion groups. Moreover, they represent multiplication by

        2

    {\displaystyle 2}

. The lines moving up and right by one unit represent multiplication by

          h

            1

    {\displaystyle h_{1}}

.

Using the spectral sequence above for X = Y = S {\displaystyle X=Y=\mathbb {S} } we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres.[2] For p = 2 {\displaystyle p=2} this amounts to looking at the E 2 {\displaystyle E_{2}} -page with

E 2 s , t = Ext A 2 s , t ( Z / 2 , Z / 2 ) {\displaystyle E_{2}^{s,t}={\text{Ext}}_{{\mathcal {A}}_{2}}^{s,t}(\mathbb {Z} /2,\mathbb {Z} /2)}

This can be done by first looking at the Adams resolution of Z / 2 {\displaystyle \mathbb {Z} /2} . Since Z / 2 {\displaystyle \mathbb {Z} /2} is in degree 0 {\displaystyle 0} , we have a surjection

A 2 ⋅ ι → Z / 2 {\displaystyle {\mathcal {A}}_{2}\cdot \iota \to \mathbb {Z} /2}

where A 2 {\displaystyle {\mathcal {A}}_{2}} has a generator in degree 0 {\displaystyle 0} denoted ι {\displaystyle \iota } . The kernel K 0 {\displaystyle K_{0}} consists of all elements S q I ι {\displaystyle Sq^{I}\iota } for admissible monomials S q I {\displaystyle Sq^{I}} generating A 2 {\displaystyle {\mathcal {A}}_{2}} , hence we have a map

⨁ I admissible A 2 ⋅ S q I ι → K 0 {\displaystyle \bigoplus _{I{\text{ admissible}}}{\mathcal {A}}_{2}\cdot Sq^{I}\iota \to K_{0}}

and we denote each of the generators mapping to S q i ι {\displaystyle Sq^{i}\iota } in the direct sum as α i {\displaystyle \alpha _{i}} , and the rest of the generators as S q I α j {\displaystyle Sq^{I}\alpha _{j}} for some j {\displaystyle j} . For example,

α 1 ↦ S q 1 ι S q 2 α 1 ↦ S q 2 , 1 ι α 2 ↦ S q 2 ι S q 1 α 2 ↦ S q 3 ι α 4 ↦ S q 4 ι S q 3 α 1 ↦ S q 3 , 1 ι α 8 ↦ S q 8 S q 2 α 2 ↦ S q 3 , 1 ι {\displaystyle {\begin{aligned}\alpha _{1}\mapsto Sq^{1}\iota &&Sq^{2}\alpha _{1}\mapsto Sq^{2,1}\iota \\\alpha _{2}\mapsto Sq^{2}\iota &&Sq^{1}\alpha _{2}\mapsto Sq^{3}\iota \\\alpha _{4}\mapsto Sq^{4}\iota &&Sq^{3}\alpha _{1}\mapsto Sq^{3,1}\iota \\\alpha _{8}\mapsto Sq^{8}&&Sq^{2}\alpha _{2}\mapsto Sq^{3,1}\iota \end{aligned}}}

Notice that the last two elements of α i {\displaystyle \alpha _{i}} map to the same element, which follows from the Adem relations. Also, there are elements in the kernel, such as S q 1 α 1 {\displaystyle Sq^{1}\alpha _{1}} since

S q 1 α 1 ↦ S q 1 S q 1 ι = 0 {\displaystyle Sq^{1}\alpha _{1}\mapsto Sq^{1}Sq^{1}\iota =0}

because of the Adem relation. We call the generator of this element in F 2 {\displaystyle F_{2}} , β 2 {\displaystyle \beta _{2}} . We can apply the same process and get a kernel K 1 {\displaystyle K_{1}} , resolve it, and so on. When we do, we get an E 1 {\displaystyle E_{1}} -page which looks like

E 1 s , t = ⋮ ⋮ ⋮ ⋮ 4 S q 4 ι , S q 3 , 1 ι S q 2 , 1 α 1 , S q 3 α 1 , S q 2 α 2 , α 4 S q 2 β 2 ⋯ 3 S q 3 ι , S q 2 , 1 ι S q 2 α 1 , S q 1 α 2 S q 1 β 2 ⋯ 2 S q 2 ι α 2 , S q 1 α 1 β 2 ⋯ 1 S q 1 ι α 1 0 ⋯ 0 ι 0 0 ⋯ 0 1 2 {\displaystyle E_{1}^{s,t}={\begin{array}{c|ccc}\vdots &\vdots &\vdots &\vdots \\4&Sq^{4}\iota ,Sq^{3,1}\iota &Sq^{2,1}\alpha _{1},Sq^{3}\alpha _{1},Sq^{2}\alpha _{2},\alpha _{4}&Sq^{2}\beta _{2}&\cdots \\3&Sq^{3}\iota ,Sq^{2,1}\iota &Sq^{2}\alpha _{1},Sq^{1}\alpha _{2}&Sq^{1}\beta _{2}&\cdots \\2&Sq^{2}\iota &\alpha _{2},Sq^{1}\alpha _{1}&\beta _{2}&\cdots \\1&Sq^{1}\iota &\alpha _{1}&0&\cdots \\0&\iota &0&0&\cdots \\\hline &0&1&2\end{array}}}

which can be expanded by computer up to degree 100 {\displaystyle 100} with relative ease. Using the found generators and relations, we can calculate the E 2 {\displaystyle E_{2}} -page with relative ease. Sometimes homotopy theorists like to rearrange these elements by having the horizontal index denote s {\displaystyle s} and the vertical index denote t − s {\displaystyle t-s} giving a different type of diagram for the E 2 {\displaystyle E_{2}} -page[2]pg 21. See the diagram above for more information.

## Generalizations

The Adams–Novikov spectral sequence is a generalization of the Adams spectral sequence introduced by [Novikov (1967)](#CITEREFNovikov1967) where ordinary cohomology is replaced by a [generalized cohomology theory](/source/Generalized_cohomology_theory), often [complex bordism](/source/Complex_bordism) or [Brown–Peterson cohomology](/source/Brown%E2%80%93Peterson_cohomology). This requires knowledge of the algebra of stable cohomology operations for the cohomology theory in question, but enables calculations which are completely intractable with the classical Adams spectral sequence.

## See also

- [Postnikov system](/source/Postnikov_system)

- [Steenrod algebra](/source/Steenrod_algebra)

- [Spectrum (topology)](/source/Spectrum_(topology))

- [Adams resolution](/source/Adams_resolution)

- [Ravenel's conjectures](/source/Ravenel's_conjectures)

## References

- [Adams, J. Frank](/source/Frank_Adams) (1958), "On the structure and applications of the Steenrod algebra", *[Commentarii Mathematici Helvetici](/source/Commentarii_Mathematici_Helvetici)*, **32** (1): 180–214, [doi](/source/Doi_(identifier)):[10.1007/BF02564578](https://doi.org/10.1007%2FBF02564578), [ISSN](/source/ISSN_(identifier)) [0010-2571](https://search.worldcat.org/issn/0010-2571), [MR](/source/MR_(identifier)) [0096219](https://mathscinet.ams.org/mathscinet-getitem?mr=0096219), [S2CID](/source/S2CID_(identifier)) [15677036](https://api.semanticscholar.org/CorpusID:15677036)

- [Adams, J. Frank](/source/Frank_Adams) (2013) [1964], [*Stable homotopy theory*](https://books.google.com/books?id=-vHtCAAAQBAJ), Lecture Notes in Mathematics, vol. 3, Springer, [ISBN](/source/ISBN_(identifier)) [9783662159422](https://en.wikipedia.org/wiki/Special:BookSources/9783662159422), [MR](/source/MR_(identifier)) [0185597](https://mathscinet.ams.org/mathscinet-getitem?mr=0185597)

- Botvinnik, Boris (1992), [*Manifolds with Singularities and the Adams–Novikov Spectral Sequence*](https://books.google.com/books?id=a3AFRbZ1JnIC), London Mathematical Society Lecture Note Series, [Cambridge University Press](/source/Cambridge_University_Press), [ISBN](/source/ISBN_(identifier)) [0-521-42608-1](https://en.wikipedia.org/wiki/Special:BookSources/0-521-42608-1)

- McCleary, John (February 2001), *A User's Guide to Spectral Sequences*, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), [Cambridge University Press](/source/Cambridge_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-521-56759-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-56759-6), [MR](/source/MR_(identifier)) [1793722](https://mathscinet.ams.org/mathscinet-getitem?mr=1793722)

- [Novikov, Sergei](/source/Sergei_Novikov_(mathematician)) (1967), "Methods of algebraic topology from the point of view of cobordism theory", *Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya* (in Russian), **31**: 855–951

- [Ravenel, Douglas C.](/source/Douglas_Ravenel) (1978), "A novice's guide to the Adams–Novikov spectral sequence", in Barratt, M. G.; Mahowald, Mark E. (eds.), *Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II*, Lecture Notes in Mathematics, vol. 658, Springer, pp. 404–475, [doi](/source/Doi_(identifier)):[10.1007/BFb0068728](https://doi.org/10.1007%2FBFb0068728), [ISBN](/source/ISBN_(identifier)) [978-3-540-08859-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-08859-2), [MR](/source/MR_(identifier)) [0513586](https://mathscinet.ams.org/mathscinet-getitem?mr=0513586)

- [Ravenel, Douglas C.](/source/Douglas_Ravenel) (2003), [*Complex cobordism and stable homotopy groups of spheres*](http://www.math.rochester.edu/people/faculty/doug/mu.html) (2nd ed.), AMS Chelsea, [ISBN](/source/ISBN_(identifier)) [978-0-8218-2967-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-2967-7), [MR](/source/MR_(identifier)) [0860042](https://mathscinet.ams.org/mathscinet-getitem?mr=0860042).

### Overviews of computations

- Isaksen, D.C.; Wang, G.; Xu, Z. (2020). "More stable stems". [arXiv](/source/ArXiv_(identifier)):[2001.04511](https://arxiv.org/abs/2001.04511) [[math.AT](https://arxiv.org/archive/math.AT)]. – computes all Adams spectral sequences for the stable homotopy groups of spheres up to degree 90

### Higher-order terms

- Baues, H.J.; Jibladze, M. (2004). "Computation of the E_3-term of the Adams spectral sequence". [arXiv](/source/ArXiv_(identifier)):[math/0407045](https://arxiv.org/abs/math/0407045).

- Baues, H.J.; Blanc, D. (2015). "Higher order derived functors and the Adams spectral sequence". *Journal of Pure and Applied Algebra*. **219** (2): 199–239. [arXiv](/source/ArXiv_(identifier)):[1108.3376](https://arxiv.org/abs/1108.3376). [doi](/source/Doi_(identifier)):[10.1016/j.jpaa.2014.04.018](https://doi.org/10.1016%2Fj.jpaa.2014.04.018). [S2CID](/source/S2CID_(identifier)) [119144480](https://api.semanticscholar.org/CorpusID:119144480).

- Baues, H.J.; Frankland, M. (2016). "2-track algebras and the Adams spectral sequence". *J. Homotopy Relat. Struct*. **11** (4): 679–713. [arXiv](/source/ArXiv_(identifier)):[1505.03885](https://arxiv.org/abs/1505.03885). [doi](/source/Doi_(identifier)):[10.1007/s40062-016-0147-x](https://doi.org/10.1007%2Fs40062-016-0147-x). [S2CID](/source/S2CID_(identifier)) [119658430](https://api.semanticscholar.org/CorpusID:119658430).

## External links

- Bruner, Robert R. (June 2, 2009), [*An Adams Spectral Sequence Primer*](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bruner-primer-2009.pdf) (PDF)

- [Hatcher, Allen](/source/Allen_Hatcher), ["The Adams Spectral Sequence"](https://pi.math.cornell.edu/~hatcher/SSAT/SSch2.pdf) (PDF), *Spectral Sequences*

## Notes

1. ^ [***a***](#cite_ref-Ravenel_1-0) [***b***](#cite_ref-Ravenel_1-1) [Ravenel, Douglas C.](/source/Douglas_Ravenel) (1986). [*Complex cobordism and stable homotopy groups of spheres*](https://web.math.rochester.edu/people/faculty/doug/mu.html). Orlando: Academic Press. [ISBN](/source/ISBN_(identifier)) [978-0-08-087440-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-087440-1). [OCLC](/source/OCLC_(identifier)) [316566772](https://search.worldcat.org/oclc/316566772).

1. ^ [***a***](#cite_ref-:0_2-0) [***b***](#cite_ref-:0_2-1) [***c***](#cite_ref-:0_2-2) Hatcher, Allen. ["Spectral Sequences in Algebraic Topology"](https://pi.math.cornell.edu/~hatcher/SSAT/SSch2.pdf) (PDF). [Archived](https://web.archive.org/web/20180728105526/http://pi.math.cornell.edu:80/~hatcher/SSAT/SSch2.pdf) (PDF) from the original on 2018-07-28.

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Adapted from the Wikipedia article [Adams spectral sequence](https://en.wikipedia.org/wiki/Adams_spectral_sequence) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Adams_spectral_sequence?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.