The Chromatic Splitting Conjecture at n=p=2

TL;DR

This paper disproves the strongest form of Hopkins' chromatic splitting conjecture at chromatic level 2 and prime 2 by showing specific non-vanishing homotopy groups that contradict the conjecture's predictions.

Contribution

It provides a counterexample at n=p=2, demonstrating that the conjectured decomposition does not hold in this case.

Findings

The homotopy group of L_1L_{K(2)}V(0) is non-zero at certain degrees.

This non-vanishing contradicts the predicted chromatic splitting decomposition.

The strongest form of the conjecture fails at n=p=2.

Abstract

We show that the strongest form of Hopkins' chromatic splitting conjecture, as stated by Hovey, cannot hold at chromatic level n=2 at the prime p=2. More precisely, for V(0) the mod 2 Moore spectrum, we prove that the kth homotopy group of L_1L_{K(2)}V(0) is not zero when k is congruent to -3 modulo 8. We explain how this contradicts the decomposition of L_1L_{K(2)}S predicted by the chromatic splitting conjecture.