Agnes Beaudry

math.AT — Algebraic Topology

Weak chromatic splitting fails at n = p for every odd prime p

Contributed by Agnes Beaudry, Tomer Schlank

We prove that for every odd prime $p$ there is a height $n$ for which the map $L_{n-1}S_p \to L_{n-1}\L_{K(n)}S_p$ is not injective on homotopy groups, hence not the inclusion of a wedge summand. In particular, for such pairs $(n,p)$ the weak chromatic splitting conjecture is false. More precisely, we prove the conjecture fails for $(n,p) = (3,3)$ and for all pairs $(n,p)$ with $p \le n < \sqrt{2}(p-1)$. In particular, the conjecture fails for $(n,p) = (p,p)$ for every odd prime $p$. This disproof was produced autonomously by Claude in a session working on Freyd's generating hypothesis with no human in the loop. This note is a human write-up of the disproof. We did not contribute any ideas.