[Paper Review] C_2-equivariant and R-motivic stable stems, II

Eva Belmont, Bertrand Guillou+1 more|arXiv (Cornell University)|Jan 7, 2020

Homotopy and Cohomology in Algebraic Topology9 references4 citations

TL;DR

This paper establishes an isomorphism between the $²$-equivariant and $²$-motivic stable homotopy groups in a range of degrees via comparison of their $²$-Bockstein spectral sequences. The key result is that Betti realization induces an isomorphism in the range $2w - s < 5$, except at $(s,w) = (0,2)$, and an injection when $2w - s = 5$, showing that $²$-motivic computations can realize $C_2$-equivariant homotopy groups in this range.

ABSTRACT

We show that the $C_2$-equivariant and $\mathbb{R}$-motivic stable homotopy groups are isomorphic in a range. This result supersedes previous work of Dugger and the third author.

Motivation & Objective

• To compare the $²$-motivic and $C_2$-equivariant stable homotopy groups in a range of degrees.
• To establish that Betti realization induces an isomorphism between these groups in a specific range of stems and weights.
• To improve upon prior work by using $²$-Bockstein spectral sequences instead of cobar complexes for a more efficient comparison.
• To show that the Adams $E_2$-pages are isomorphic in the range $2w - s < 5$, except at $(s,w) = (0,2)$, and to extend this to higher Adams pages and stable homotopy groups.
• To demonstrate the sharpness of the range by constructing explicit elements outside the isomorphism range that are permanent cycles but not isomorphic under Betti realization.

Proposed method

• Compare the $²$-Bockstein spectral sequences converging to the $²$-motivic and $C_2$-equivariant Adams $E_2$-pages.
• Use the fact that the $²$-Bockstein spectral sequences are isomorphic in a range, implying isomorphism of the Adams $E_2$-pages.
• Apply induction on the Adams spectral sequence pages $E_r$ to extend the isomorphism from $E_2$ to $E_∞$.
• Leverage the vanishing of the cokernel of the Betti realization map, specifically $\operatorname{Ext}_{NC}$, to prove isomorphism in the stated range.
• Use the fact that $\frac{\gamma}{\tau}h_0^f$ is a permanent cycle in the $C_2$-equivariant Adams spectral sequence to handle edge cases in the induction.
• Use diagram chasing on the Adams differential diagrams to show that the Betti realization map remains an isomorphism or injection through higher pages.

Experimental results

Research questions

• RQ1 In which range of stems and weights are the $²$-motivic and $C_2$-equivariant stable homotopy groups isomorphic under Betti realization?
• RQ2 Can the comparison of Adams $E_2$-pages be extended to higher Adams pages and ultimately to the stable homotopy groups?
• RQ3 Is the range $2w - s < 5$ sharp for the isomorphism of stable homotopy groups under Betti realization?
• RQ4 What is the behavior of the Betti realization map when $2w - s = 5$, and why is it only injective there?
• RQ5 Are there explicit permanent cycles outside the isomorphism range that demonstrate the sharpness of the bound?

Key findings

• The Betti realization map $\pi^\mathbb{R}_{s,w} \to \pi^{C_2}_{s,w}$ is an isomorphism when $2w - s < 5$ and $(s,w) \neq (0,2)$.
• The map is an injection when $2w - s = 5$, and this range is sharp, as shown by explicit counterexamples.
• The $²$-Bockstein spectral sequences for the $²$-motivic and $C_2$-equivariant Adams $E_2$-pages are isomorphic in the range $2w - s < 5$, except at $(0,2)$.
• The Adams $E_\infty$-pages are isomorphic in the range $2w - s < 5$, $(s,w) \neq (0,2)$, due to isomorphic associated graded objects.
• The element $\frac{\gamma}{\tau}P^k h_1$ in degree $(8k+1, 4k+1, 4k+3)$ is a permanent cycle in both spectral sequences and lies outside the isomorphism range, showing that the bound is sharp.
• The map fails to be an isomorphism at $2w - s = 5$ for $k \geq 1$, as these elements are not isomorphic under Betti realization, confirming the sharpness of the range.