The Chromatic Splitting Conjecture 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.

Revised version. To appear in Geometry & Topology

References in corpus (3)

• Towards the homotopy of the $K(2)$-local Moore spectrum at $p=2$
• The Algebraic Duality Resolution at $p=2$
• The $E_2$-term of the $K(n)$-local $E_n$-Adams spectral sequence

Cited by in corpus (6)

• Topological resolutions in K(2)-local homotopy theory at the prime 2
• Monochromatic homotopy theory is asymptotically algebraic
• Chromatic splitting for the $K(2)$-local sphere at $p=2$
• Algebraic chromatic homotopy theory for $B{P}_{\ast }BP$-comodules
• The algebraic chromatic splitting conjecture for Noetherian ring spectra
• Morava K-theory and Filtrations by Powers