Adams spectral sequence

Classical formulation

Formulation for computing homotopy groups of spectra

The classical Adams spectral sequence can be stated for any connective spectrum ${\displaystyle X}$ of finite type, meaning ${\displaystyle {\pi }_i(X)=0}$ for ${\displaystyle i<0}$ and ${\displaystyle {\pi }_i(X)}$ is a finitely generated Abelian group in each degree. Then, there is a spectral sequence ${\displaystyle E_{\ast }^{\ast ,\ast }(X)}$ such that

1. ${\displaystyle E_2^{s,t}={\text{Ext}}_{A_p}^{s,t}(H^{\ast }(X),\mathbb{Z}/p)}$ for ${\displaystyle A_p}$ the mod ${\displaystyle p}$ Steenrod algebra
2. For ${\displaystyle X}$ of finite type, ${\displaystyle E_{\mathrm{\infty }}^{\ast ,\ast }}$ is a bigraded group associated with a filtration of ${\displaystyle {\pi }_{\ast }(X)\otimes {\mathbb{Z}}_p}$ (the p-adic integers)

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

This statement generalizes a little bit further by replacing the ${\displaystyle {\mathcal{A}}_p}$-module ${\displaystyle \mathbb{Z}/p}$ with the cohomology groups ${\displaystyle H^{\ast }(Y)}$ for some connective spectrum ${\displaystyle Y}$ (or topological space ${\displaystyle Y}$). This is because the construction of the spectral sequence uses a "free" resolution of ${\displaystyle H^{\ast }(X)}$ as an ${\displaystyle {\mathcal{A}}_p}$-module, hence we can compute the Ext groups with ${\displaystyle H^{\ast }(Y)}$ as the second entry. We therefore get a spectral sequence with ${\displaystyle E_2}$-page given by

${\displaystyle E_2^{t,s}={\text{Ext}}_{{\mathcal{A}}_p}^{s,t}(H^{\ast }(X),H^{\ast }(Y))}$

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

${\displaystyle E_2^{s,t}\Rightarrow {\pi }_{t-s}^{\mathbb{S}}([X,Y])\otimes {\mathbb{Z}}_p}$

Spectral sequence for the stable homotopy groups of spheres

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

${\displaystyle E_2^{t,s}={\text{Ext}}_{{\mathcal{A}}_p}^{s,t}(H^{\ast }(\mathbb{S}),H^{\ast }(\mathbb{S}))\Rightarrow {\pi }_{t-s}(\mathbb{S})\otimes {\mathbb{Z}}_p}$

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^{pp 23–25}. Also note that we can rewrite ${\displaystyle H^{\ast }(\mathbb{S})=\mathbb{Z}/p}$, so the ${\displaystyle E_2}$-page is

${\displaystyle E_2^{t,s}={\mathrm{Ext}}_{{\mathcal{A}}_p}^{s,t}(\mathbb{Z}/p,\mathbb{Z}/p)\Rightarrow {\pi }_{t-s}(\mathbb{S})\otimes {\mathbb{Z}}_p}$

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

Ext terms from the resolution

Given the Adams resolution

${\displaystyle \cdots \to H^{\ast }(F_2)\to H^{\ast }(F_1)\to H^{\ast }(F_0)\to H^{\ast }(X)}$

we have the ${\displaystyle E_1}$-terms as

${\displaystyle E_1^{s,t}={\mathrm{Hom}}_{{\mathcal{A}}_p}^t(H^{\ast }(F_s),H^{\ast }(Y))}$

for the graded Hom-groups. Then the ${\displaystyle E_1}$-page can be written as

${\displaystyle E_1=\begin{array}{cccc}3& ⋮& ⋮& ⋮\\ 2& {\text{Hom}}^2(H^{\ast }(F_0),H^{\ast }(Y))& {\text{Hom}}^2(H^{\ast }(F_1),H^{\ast }(Y))& {\text{Hom}}^2(H^{\ast }(F_2),H^{\ast }(Y))& \cdots \\ 1& {\text{Hom}}^1(H^{\ast }(F_0),H^{\ast }(Y))& {\text{Hom}}^1(H^{\ast }(F_1),H^{\ast }(Y))& {\text{Hom}}^1(H^{\ast }(F_2),H^{\ast }(Y))& \cdots \\ 0& {\text{Hom}}^0(H^{\ast }(F_0),H^{\ast }(Y))& {\text{Hom}}^0(H^{\ast }(F_1),H^{\ast }(Y))& {\text{Hom}}^0(H^{\ast }(F_2),H^{\ast }(Y))& \cdots \\ & 0& 1& 2\end{array}}$

so the degree of ${\displaystyle s}$ 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 ${\displaystyle r}$th Adams differential always goes to the left 1, and up ${\displaystyle r}$. That is,

${\displaystyle d_r:E_r^{s,t}\to E_r^{s-1,t+r}}$.

Examples with Eilenberg–Maclane spectra

Some of the simplest calculations are with Eilenberg–Maclane spectra such as ${\displaystyle X=H\mathbb{Z}}$ and ${\displaystyle X=H\mathbb{Z}/(p^k)}$. For the first case, we have the ${\displaystyle E_1}$ page

${\displaystyle E_1^{s,t}=\{\begin{array}{ll}\mathbb{Z}/p& \text{ if }t=s\\ 0& \text{ otherwise }\end{array}}$

giving a collapsed spectral sequence, hence ${\displaystyle E_1=E_{\mathrm{\infty }}}$. This can be rewritten as

${\displaystyle {\text{Ext}}_{{\mathcal{A}}_p}^{s,t}(H^{\ast }(H\mathbb{Z}),\mathbb{Z}/p)=\{\begin{array}{ll}\mathbb{Z}/p& \text{ if }t=s\\ 0& \text{ if }t\ne s\end{array}}$

giving the ${\displaystyle E_2}$-page. For the other case, note there is a cofiber sequence

${\displaystyle H\mathbb{Z}\stackrel{\cdot p^k}{\to }H\mathbb{Z}\to H\mathbb{Z}/p^k\to \mathrm{\Sigma }H\mathbb{Z}}$

which ends up giving a splitting in cohomology, so ${\displaystyle H^{\ast }(H\mathbb{Z}/p^k)=H^{\ast }(H\mathbb{Z})\oplus H^{\ast }(\mathrm{\Sigma }H\mathbb{Z})}$ as ${\displaystyle {\mathcal{A}}_p}$-modules. Then, the ${\displaystyle E_2}$-page of ${\displaystyle H^{\ast }(H\mathbb{Z}/p)}$ can be read as

${\displaystyle E_2^{s,t}=\{\begin{array}{ll}\mathbb{Z}/p& \text{if }t-s=0,1\\ 0& \text{otherwise }\end{array}}$

The expected ${\displaystyle E_{\mathrm{\infty }}}$-page is

${\displaystyle E_{\mathrm{\infty }}^{s,t}=\{\begin{array}{ll}\mathbb{Z}/p^k& \text{ if }t=s\\ 0& \text{ otherwise }\end{array}}$.

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 ${\displaystyle (s,s+1)}$.

Other applications

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

The Thom isomorphism theorem relates differential topology to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960, John Milnor and Sergei Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism. Further, Milnor and C. T. C. Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented cobordism ring: two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel–Whitney numbers agree.

Stable homotopy groups of spheres

Using the spectral sequence above for ${\displaystyle X=Y=\mathbb{S}}$ we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres. For ${\displaystyle p=2}$ this amounts to looking at the ${\displaystyle E_2}$-page with

${\displaystyle E_2^{s,t}={\text{Ext}}_{{\mathcal{A}}_2}^{s,t}(\mathbb{Z}/2,\mathbb{Z}/2)}$

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

${\displaystyle {\mathcal{A}}_2\cdot \iota \to \mathbb{Z}/2}$

where ${\displaystyle {\mathcal{A}}_2}$ has a generator in degree ${\displaystyle 0}$ denoted ${\displaystyle \iota }$. The kernel ${\displaystyle K_0}$ consists of all elements ${\displaystyle S{q}^I\iota }$ for admissible monomials ${\displaystyle S{q}^I}$ generating ${\displaystyle {\mathcal{A}}_2}$, hence we have a map

${\displaystyle \underset{I\text{ admissible}}{⨁}{\mathcal{A}}_2\cdot S{q}^I\iota \to K_0}$

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

${\displaystyle \begin{array}{ccc}{\alpha }_1\mapsto S{q}^1\iota & & S{q}^2{\alpha }_1\mapsto S{q}^{2,1}\iota \\ {\alpha }_2\mapsto S{q}^2\iota & & S{q}^1{\alpha }_2\mapsto S{q}^3\iota \\ {\alpha }_4\mapsto S{q}^4\iota & & S{q}^3{\alpha }_1\mapsto S{q}^{3,1}\iota \\ {\alpha }_8\mapsto S{q}^8& & S{q}^2{\alpha }_2\mapsto S{q}^{3,1}\iota \end{array}}$

Notice that the last two elements of ${\displaystyle {\alpha }_i}$ map to the same element, which follows from the Adem relations. Also, there are elements in the kernel, such as ${\displaystyle S{q}^1{\alpha }_1}$ since

${\displaystyle S{q}^1{\alpha }_1\mapsto S{q}^{1}S{q}^1\iota =0}$

because of the Adem relation. We call the generator of this element in ${\displaystyle F_2}$, ${\displaystyle {\beta }_2}$. We can apply the same process and get a kernel ${\displaystyle K_1}$, resolve it, and so on. When we do, we get an ${\displaystyle E_1}$-page which looks like

${\displaystyle E_1^{s,t}=\begin{array}{cccc}⋮& ⋮& ⋮& ⋮\\ 4& S{q}^4\iota ,S{q}^{3,1}\iota & S{q}^{2,1}{\alpha }_1,S{q}^3{\alpha }_1,S{q}^2{\alpha }_2,{\alpha }_4& S{q}^2{\beta }_2& \cdots \\ 3& S{q}^3\iota ,S{q}^{2,1}\iota & S{q}^2{\alpha }_1,S{q}^1{\alpha }_2& S{q}^1{\beta }_2& \cdots \\ 2& S{q}^2\iota & {\alpha }_2,S{q}^1{\alpha }_1& {\beta }_2& \cdots \\ 1& S{q}^1\iota & {\alpha }_1& 0& \cdots \\ 0& \iota & 0& 0& \cdots \\ & 0& 1& 2\end{array}}$

which can be expanded by computer up to degree ${\displaystyle 100}$ with relative ease. Using the found generators and relations, we can calculate the ${\displaystyle E_2}$-page with relative ease. Sometimes homotopy theorists like to rearrange these elements by having the horizontal index denote ${\displaystyle s}$ and the vertical index denote ${\displaystyle t-s}$ giving a different type of diagram for the ${\displaystyle E_2}$-page^{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) where ordinary cohomology is replaced by a generalized cohomology theory, often complex bordism or Brown–Peterson 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.