Adams spectral sequence

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 ${\displaystyle E_{*}^{*,*}(X)}$[1]^: 41 such that

1. ${\displaystyle E_{2}^{s,t}={\text{Ext}}_{A_{p}}^{s,t}(H^{*}(X),\mathbb {Z} /p)}$ for ${\displaystyle A_{p}}$ the mod ${\displaystyle p}$ Steenrod algebra
2. For ${\displaystyle X}$ of finite type, ${\displaystyle E_{\infty }^{*,*}}$ is a bigraded group associated with a filtration of ${\displaystyle \pi _{*}(X)\otimes \mathbb {Z} _{p}}$ (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 ${\displaystyle \Sigma ^{\infty }Y}$, 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 ${\displaystyle H^{*}(Y)}$ 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 ${\displaystyle H^{*}(X)}$ as an ${\displaystyle {\mathcal {A}}_{p}}$-module, hence we can compute the Ext groups with ${\displaystyle H^{*}(Y)}$ as the second entry. We therefore get a spectral sequence with ${\displaystyle E_{2}}$-page given by

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

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

${\displaystyle E_{2}^{s,t}\Rightarrow \pi _{k}^{\mathbb {S} }([X,Y])\otimes \mathbb {Z} _{p}}$

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

${\displaystyle E_{2}^{t,s}={\text{Ext}}_{{\mathcal {A}}_{p}}^{s,t}(H^{*}(\mathbb {S} ),H^{*}(\mathbb {S} ))\Rightarrow \pi _{*}(\mathbb {S} )\otimes \mathbb {Z} _{p}}$

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 ${\displaystyle H^{*}(\mathbb {S} )=\mathbb {Z} /p}$, so the ${\displaystyle E_{2}}$-page is

${\displaystyle E_{2}^{t,s}=\operatorname {Ext} _{{\mathcal {A}}_{p}}^{s,t}(\mathbb {Z} /p,\mathbb {Z} /p)\Rightarrow \pi _{*}(\mathbb {S} )\otimes \mathbb {Z} _{p}}$

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

Given the Adams resolution

${\displaystyle \cdots \to H^{*}(F_{2})\to H^{*}(F_{1})\to H^{*}(F_{0})\to H^{*}(X)}$

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

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

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

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

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

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

${\displaystyle d_{r}\colon E_{r}^{s,t}\to E_{r}^{s+r,t+r-1}}$.

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

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

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

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

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

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

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

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

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

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

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

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.