Motivic and Slice Spectral Sequences¶
Job: develop the motivic Adams, the slice spectral sequence (Voevodsky), the motivic Adams–Novikov / MGL, and the \(\mathbb{R}\)-motivic / \(C_2\)-equivariant — with the motivic Adams \(E_2\) / \(\eta\) as the worked example.
Core¶
The motivic Adams spectral sequence¶
| Motivic Adams / \(\mathcal{A}^{\mathrm{mot}}\) / \(E_2\) | Description |
|---|---|
| Motivic Adams | \(E_2=\mathrm{Ext}_{\mathcal{A}^{\mathrm{mot}}}^{s,t,w}(\mathbb{F}_2,H^{*,*}(X))\Rightarrow\pi_{t-s,w}^{\hat{}}(X)\) (the motivic Steenrod, the motivic homotopy / motivic Adams) |
| Motivic stems | \(\pi_{*,*}^{\mathrm{mot}}\mathbb{1}\) the motivic stable stems (the motivic homotopy / motivic Adams) |
The slice spectral sequence (Voevodsky)¶
| Slice / \(s_q\) / \(E^1\) | Description |
|---|---|
| Slice SS | \(E^1_{p,q}=H^{p-q}(s_q E)\Rightarrow H^{p+q}(E)\) (the slices \(s_q E\), the motivic homotopy / slice filtration) |
| Motivic Postnikov | the slice the motivic Postnikov (the motivic homotopy / slice filtration) |
The motivic Adams–Novikov and the MGL¶
| Motivic Adams–Novikov / MGL / \(E_2\) | Description |
|---|---|
| MGL-based | the \(\mathrm{MGL}\) the algebraic cobordism (the motivic homotopy / algebraic cobordism) |
| Motivic Adams–Novikov | the \(\mathrm{MGL}\) the Adams–Novikov (the motivic homotopy / algebraic cobordism) |
The \(\mathbb{R}\)-motivic and the \(C_2\)-equivariant¶
| \(\mathbb{R}\)-motivic / \(C_2\) / Kervaire / HHR | Description |
|---|---|
| \(\mathbb{R}\)-motivic | the motivic homotopy / motivic Adams, the \(C_2\)-equivariant |
| Kervaire / HHR | the Kervaire invariant, the Hill–Hopkins–Ravenel (the motivic homotopy / motivic Adams) |
See it¶
Motivic Adams \(E_2\) for \(\pi_{*,*}^{\mathrm{mot}}\mathbb{1}\) over \(\mathbb{C}\) and the \(\eta\). The motivic Adams: \(E_2=\mathrm{Ext}_{\mathcal{A}^{\mathrm{mot}}}^{s,t,w}(\mathbb{F}_2,\mathbb{F}_2)\Rightarrow\pi_{t-s,w}^{\hat{}}\mathbb{1}\) (the \(\mathcal{A}^{\mathrm{mot}}\) the motivic Steenrod). The \(\eta\): \(\pi_{1,1}^{\mathrm{mot}}\mathbb{1}=\mathbb{Z}/2\) the motivic Hopf (the \(E_2^{1,1,1}=\mathbb{Z}/2\) the \(h_{1,1}\), the motivic homotopy / motivic Adams). The key point: the motivic Adams the \(E_2\) the \(\mathcal{A}^{\mathrm{mot}}\), the \(\eta\) the motivic Hopf — the motivic stable stems.
Check¶
Show that the motivic Adams \(E_2\) computes \(\pi_{*,*}^{\mathrm{mot}}\mathbb{1}\), and the \(\eta\).
The motivic Adams: the \(E_2=\mathrm{Ext}_{\mathcal{A}^{\mathrm{mot}}}^{s,t,w}(\mathbb{F}_2,\mathbb{F}_2)\) (the \(\mathcal{A}^{\mathrm{mot}}\) the Voevodsky motivic Steenrod, the motivic homotopy / motivic Adams). The \(\eta\): the \(\pi_{1,1}^{\mathrm{mot}}\mathbb{1}=\mathbb{Z}/2\) the motivic Hopf map (the \(h_{1,1}\) the \(E_2^{1,1,1}\), the Morel \(\pi_{n,n}^{\mathrm{mot}}\mathbb{1}=\mathbf{K}^{\mathrm{MW}}_n\)). The Betti: the \(\pi_{n,0}^{\mathrm{mot}}\to\pi_n^{\mathrm{top}}\) the realization. So the motivic Adams the \(\mathcal{A}^{\mathrm{mot}}\), the \(\eta\) the motivic Hopf.
Explain the slice spectral sequence and the Voevodsky slice filtration.
The slice SS: the \(E^1_{p,q}=H^{p-q}(s_q E)\Rightarrow H^{p+q}(E)\) (the \(s_q E\) the slices, the motivic homotopy / slice filtration). The slice filtration: the Voevodsky tower \(\cdots\to f_{q+1}E\to f_qE\to\cdots\) (the motivic Postnikov, the \(s_qE=f_qE/f_{q+1}E\) the slice). The \(s_0\mathbb{1}=\mathbf{H}\mathbb{Z}\) (the Voevodsky, the motivic homotopy / slice filtration). The convergence: the Hayashi the connective \(E\) the convergent. So the slice SS the Voevodsky, the \(s_q\) the slice.
Pitfalls¶
- The motivic Adams the \(\mathcal{A}^{\mathrm{mot}}\) (the Voevodsky motivic Steenrod, the motivic homotopy / motivic Adams) — the \(\mathcal{A}^{\mathrm{mot}}\) the motivic, the \(E_2\) the \(\mathrm{Ext}\).
- The slice the Voevodsky (the \(s_q\), the motivic homotopy / slice filtration) — the slice the motivic Postnikov, the \(s_0\mathbb{1}=\mathbf{H}\mathbb{Z}\).
- The MGL the Adams–Novikov (the motivic homotopy / algebraic cobordism) — the \(\mathrm{MGL}\) the algebraic cobordism, the motivic Adams–Novikov.
- The \(\mathbb{R}\)-motivic / \(C_2\) (the Kervaire, the HHR, the motivic homotopy / motivic Adams) — the \(\mathbb{R}\)-motivic the \(C_2\), the Kervaire the HHR.
Refs¶
- Morel, "\(\mathbb{A}^1\)-algebraic topology" (2006); the motivic Adams / the \(\eta\)
- Voevodsky, "A possible new approach to the motivic stable homotopy" (2002); the slice filtration
- Hill–Hopkins–Ravenel, "On the non-existence of Kervaire invariant one" (2016); the \(C_2\)-equivariant
- Isaksen–Wang–Xu, "Stable homotopy groups of spheres" (2023); the \(\mathbb{R}\)-motivic
Next¶
SS index · Foundations · Adams & Adams–Novikov · Eilenberg–Moore & Bar · Bockstein & Descent · Motivic homotopy / motivic Adams · Motivic homotopy / slice filtration