Adams and Adams–Novikov Spectral Sequences¶

Job: develop the Adams spectral sequence (Steenrod algebra / stable stems), the Adams–Novikov (BP / chromatic), the May spectral sequence (Steenrod algebra Ext), and the chromatic spectral sequence ($v_n$-periodic / $K(n)$-local) — with the Adams $E_2$ for $\pi_*^{\mathrm{top}}\mathbb{S}$ and the $\eta$ / Hopf as the worked example.

Core¶

The Adams spectral sequence¶

The Adams spectral sequence: $$ E_2^{s,t}=\mathrm{Ext}{\mathcal{A}}^{s,t}(\mathbb{F}_2,H^*(X))\Rightarrow\pi(X), $$ the Steenrod algebra }^{\hat{}$\mathcal{A}$, the stable homotopy, the topology / algebraic topology. At odd primes, replace $\mathbb{F}_2$ with $\mathbb{F}_p$ and the mod-$p$ Steenrod algebra.

Adams	Description
$E_2=\mathrm{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_2,H^*(X))$	the Steenrod algebra $\mathcal{A}$, the $\mathrm{Ext}$ over $\mathcal{A}$
Converges to $\pi_{t-s}^{\hat{}}(X)$	the $2$-complete stable homotopy (the $\hat{}$ the $2$-adic completion)
Stable stems	$\pi_*^{\mathrm{top}}\mathbb{S}$ the stable homotopy groups of spheres (the Adams the computation)
$\mathcal{A}$	the Steenrod algebra (the $Sq^i$ at $p=2$, the $P^i$ at odd $p$, the mod-$p$ cohomology operations)

The Adams–Novikov spectral sequence¶

The Adams–Novikov spectral sequence: $$ E_2^{s,t}=\mathrm{Ext}{\mathrm{BP}}^{s,t}(\mathrm{BP}_,\mathrm{BP}*(X))\Rightarrow\pi(X), $$ the Brown–Peterson }^{\hat{}$\mathrm{BP}$ (the $p$-local summand of $\mathrm{MU}$), the $\mathrm{MU}$-based, the topology.

Adams–Novikov	Description
$E_2=\mathrm{Ext}_{\mathrm{BP}_}(\mathrm{BP}_,\mathrm{BP}_*(X))$	the $\mathrm{BP}$ the Brown–Peterson (the $p$-local $\mathrm{MU}$), the $\mathrm{Ext}$ over $\mathrm{BP}_*$
Chromatic	the chromatic filtration (the $v_n$-periodic, the motivic homotopy / slice)
$v_n$-periodic	the $v_n$ self-map, the $K(n)$-local, the chromatic layers
$\mathrm{BP}_*$	the $\mathrm{BP}$ coefficient ring $\mathbb{Z}_{(p)}[v_1,v_2,\dots]$ (the Hazewinkel generators)

The May spectral sequence and the Steenrod algebra¶

May / Steenrod / Milnor	Description
May spectral sequence	$\mathrm{Ext}_{\mathcal{A}}$ via the May SS (the associated graded of $\mathcal{A}$, the polynomial algebra, the topology)
Steenrod / Milnor	$\mathcal{A}$ the Milnor basis (the dual $\mathcal{A}_*$ the polynomial, the topology)
$\mathcal{A}_*$	the dual Steenrod algebra (the polynomial $\mathbb{F}_2[\xi_1,\xi_2,\dots]$, the Milnor)

The chromatic spectral sequence and the $v_n$-periodic¶

Chromatic / $v_n$ / $K(n)$	Description
Chromatic spectral sequence	the $v_n$-periodic, the $K(n)$-local, the chromatic resolution (the motivic homotopy / motivic Adams)
$v_n$-periodic	the $v_n$ self-map (the Smith–Toda, the $v_n$-periodic spectra)
$K(n)$-local	the $K(n)$-local stable homotopy (the Bousfield localization, the Morava $K$-theory)
Stable stems	the chromatic the resolution of the stable stems (the motivic homotopy / motivic Adams)

See it¶

Adams $E_2$ for $\pi_*^{\mathrm{top}}\mathbb{S}$ and the $\eta$ / the Hopf. The Adams: $E_2^{s,t}=\mathrm{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_2,\mathbb{F}_2)\Rightarrow\pi_{t-s}^{\hat{}}\mathbb{S}$ (the Steenrod $\mathcal{A}$, the $\mathbb{F}_2$). The $\eta$: the $\pi_1^{\mathrm{top}}\mathbb{S}=\mathbb{Z}/2$ the Hopf map $\eta$ (the $E_2^{1,2}=\mathbb{Z}/2$ the $h_0$ the $\eta$, the $h_0\in\mathrm{Ext}_{\mathcal{A}}^{1,2}$ the generator, the Adams the computation). The $\nu$: the $\pi_3^{\mathrm{top}}\mathbb{S}=\mathbb{Z}/24$ (the $h_1\in E_2^{1,4}$ the $\nu$, the $h_1^3=0$ the relation). The key point: the Adams the $E_2$ the Steenrod, the $\eta$ the Hopf — the stable stems the Adams.

Check¶

Show that the Adams $E_2^{1,2}=\mathbb{Z}/2$ detects the Hopf map $\eta\in\pi_1^{\mathrm{top}}\mathbb{S}$, and explain the $h_0$.

The Adams $E_2^{s,t}=\mathrm{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_2,\mathbb{F}_2)$ (the $\mathrm{Ext}$ over the Steenrod algebra $\mathcal{A}$, the $\mathbb{F}_2$ the trivial module). The $s=1,t=2$: the $\mathrm{Ext}_{\mathcal{A}}^{1,2}(\mathbb{F}_2,\mathbb{F}_2)$ — the $1$-extensions of $\mathbb{F}_2$ by $\mathbb{F}_2$ in degree $2$ (the $\mathrm{Ext}^1$, the short exact $0\to\mathbb{F}_2\to M\to\mathbb{F}_2\to0$ the degree shift). The $h_0$: the generator of $\mathrm{Ext}_{\mathcal{A}}^{1,2}$ (the $Sq^2$ the operation, the $h_0$ the dual, the $1$-extension the $Sq^2$). The $\eta$: the $\pi_1^{\mathrm{top}}\mathbb{S}=\mathbb{Z}/2$ the Hopf map $\eta$ (the $S^3\to S^2$ the Hopf fibration, the suspension the stable $\eta:S^1\to S^0$). The Adams: the $h_0$ the $E_2^{1,2}$ detects the $\eta$ (the $d_r$ the no-differential, the $E_\infty^{1,2}=\mathbb{Z}/2$ the $\eta$). So the $h_0$ the $\mathrm{Ext}^{1,2}$ the Hopf $\eta$.

Explain the Adams–Novikov $E_2$ via $\mathrm{BP}_*$ and the chromatic connection, and the relation to the Adams.

The Adams–Novikov: $E_2=\mathrm{Ext}_{\mathrm{BP}_*}(\mathrm{BP}_*,\mathrm{BP}_*(X))$ (the $\mathrm{BP}$ the Brown–Peterson, the $p$-local $\mathrm{MU}$, the $\mathrm{Ext}$ over $\mathrm{BP}_*$). The $\mathrm{BP}_*=\mathbb{Z}_{(p)}[v_1,v_2,\dots]$ (the Hazewinkel, the $|v_n|=2(p^n-1)$). The $\mathrm{Ext}_{\mathrm{BP}_*}$: the $\mathrm{Ext}$ over the polynomial ring $\mathrm{BP}_*$ (the $\mathrm{BP}_*\mathrm{BP}$ the Hopf algebroid, the Adams–Novikov the $E_2$ the $\mathrm{Ext}$ over the Hopf algebroid). The chromatic: the $v_n$-periodic (the $v_n$ the generator, the $K(n)$-local, the chromatic the resolution), the chromatic the filtration (the $v_n$-periodic the layers). The relation to the Adams: the Adams the Steenrod $\mathcal{A}$ (the mod-$p$), the Adams–Novikov the $\mathrm{BP}$ (the $p$-local $\mathrm{MU}$, the richer), the Adams–Novikov the finer (the chromatic the $v_n$, the Adams the cruder). The chromatic: the Adams–Novikov the chromatic (the $v_n$-periodic), the Adams the Steenrod (the mod-$p$), the Adams–Novikov the chromatic / the finer.

Pitfalls¶

The Adams the $\ell$-complete (the $\pi_*^{\hat{}}$ the $\ell$-adic completion, not the integral) — the Adams the $\ell$-complete stable stems, the integral the extension (the arithmetic fracture the gluing).
The Adams–Novikov the $\mathrm{BP}$ (the $p$-local, the chromatic — the Adams the Steenrod, the Adams–Novikov the $\mathrm{BP}$, the finer) — the Adams–Novikov the chromatic, the Adams the Steenrod.
The chromatic the $v_n$-periodic (the $K(n)$-local, the Bousfield — the $v_n$ the self-map, the $K(n)$ the localization) — the chromatic the layers, the $v_n$ the periodic.
The May SS the $\mathrm{Ext}_{\mathcal{A}}$ via the associated graded (the polynomial, the May the tool) — the May the computation of $\mathrm{Ext}_{\mathcal{A}}$, the Steenrod the algebra.

Refs¶

Adams, "On the structure and applications of the Steenrod algebra" (1958); the Adams spectral sequence
Novikov, "The methods of algebraic topology from the viewpoint of cobordism theory" (1967); the Adams–Novikov
May, "The cohomology of restricted Lie algebras and of Hopf algebras" (1966); the May SS
Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres (the chromatic, the $v_n$-periodic)

Next¶

SS index · Foundations · Eilenberg–Moore & Bar · Motivic & Slice · Bockstein & Descent · Topology / algebraic topology · Motivic homotopy / motivic Adams · Motivic homotopy / slice

Adams	Description
\(E_2=\mathrm{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_2,H^*(X))\)	the Steenrod algebra \(\mathcal{A}\), the \(\mathrm{Ext}\) over \(\mathcal{A}\)
Converges to \(\pi_{t-s}^{\hat{}}(X)\)	the \(2\)-complete stable homotopy (the \(\hat{}\) the \(2\)-adic completion)
Stable stems	\(\pi_*^{\mathrm{top}}\mathbb{S}\) the stable homotopy groups of spheres (the Adams the computation)
\(\mathcal{A}\)	the Steenrod algebra (the \(Sq^i\) at \(p=2\), the \(P^i\) at odd \(p\), the mod-\(p\) cohomology operations)

Adams–Novikov	Description
\(E_2=\mathrm{Ext}_{\mathrm{BP}_}(\mathrm{BP}_,\mathrm{BP}_*(X))\)	the \(\mathrm{BP}\) the Brown–Peterson (the \(p\)-local \(\mathrm{MU}\)), the \(\mathrm{Ext}\) over \(\mathrm{BP}_*\)
Chromatic	the chromatic filtration (the \(v_n\)-periodic, the motivic homotopy / slice)
\(v_n\)-periodic	the \(v_n\) self-map, the \(K(n)\)-local, the chromatic layers
\(\mathrm{BP}_*\)	the \(\mathrm{BP}\) coefficient ring \(\mathbb{Z}_{(p)}[v_1,v_2,\dots]\) (the Hazewinkel generators)

Chromatic / \(v_n\) / \(K(n)\)	Description
Chromatic spectral sequence	the \(v_n\)-periodic, the \(K(n)\)-local, the chromatic resolution (the motivic homotopy / motivic Adams)
\(v_n\)-periodic	the \(v_n\) self-map (the Smith–Toda, the \(v_n\)-periodic spectra)
\(K(n)\)-local	the \(K(n)\)-local stable homotopy (the Bousfield localization, the Morava \(K\)-theory)
Stable stems	the chromatic the resolution of the stable stems (the motivic homotopy / motivic Adams)