Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms)

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Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms) evidence for this event

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A rational obstruction to strong chromatic splitting at height three evidence for this event

Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture (1995) evidence for this event

Failure of finite assembly for a chromatic overlap at the prime three evidence for this event

VibeMathed record: Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms) evidence for this event

Weak form: for the derived $p$-completion $X=S_p^\wedge$, the canonical map $L_{n-1}X\to L_{n-1}L_{K(n)}X$ has no homotopy retraction at height $n=p$ for every $p\ge5$ and at $n=p+1$ for every $p\ge7$; the class $\beta_1^{(p-1)^2}$ is a nonzero kernel element (at $p=5$, $\beta_1^{16}$ in degree 608). Strong form: for $p\ge5$ the map $L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ is nonzero on $\pi_{-3}$, so the height-three splitting fails even as an equivalence of $E(2)$-local spectra. Companions: at $n=p=3$ the overlap is not in the thick subcategory generated by $L_0S,L_1S,L_2S$; positively, for $p>n+1$ the overlap has a $2^n$-stage filtration with the predicted cofibers (nonsplit), with explicit attachments at height three. Heights below $p-1$ remain open. parent of this event

The height-three chromatic overlap: an explicit filtration and its attachments evidence for this event

Filtered chromatic splitting at generic primes evidence for this event

Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms) parent of this event