Chromatic splitting: filtrations and counterexamples

Let n\geq1 and let p>n+1 be prime. There exist an enumeration I_1,\ldots,I_{2^n} of the subsets of \{1,\ldots,n\}, a filtration F_\bullet, and a family of classes y_i\in\pi_{1-2i}D, 1\leq i\leq n, with the following properties.

1. (a)

   Filtration and unit. The enumeration has I_1=\varnothing and \max I_j nondecreasing for j\geq2. The filtration is

   0=F_0\longrightarrow F_1\longrightarrow\cdots \longrightarrow F_{2^n}\xrightarrow{\simeq}X_{n,p}

   with \operatorname{cofib}(F_{j-1}\to F_j)\simeq C_{I_j}, where

   \begin{aligned} C_\varnothing&=L_{n-1}S,\\ C_I&=\Sigma^{-d(I)}L_{n-\max I}S \quad(I\ne\varnothing). \end{aligned}

   Under F_1=C_\varnothing, the composite F_1\to X_{n,p} is the canonical localization unit.

2. (b)

   Simultaneous product bases. For this one family, set y_\varnothing=1 and y_I=y_{i_1}\cdots y_{i_r} for I=\{i_1<\cdots<i_r\}. For every 1\leq t<n, the specified map

   L_{K(t)} \left(\bigvee_{I\subseteq\{1,\ldots,n-t\}} \Sigma^{-d(I)}S\xrightarrow{(y_I)}D\right) \quad\text{is an equivalence}.

For every prime p\geq5, there is no equivalence of E(2)-local spectra L_2L_{K(3)}S\simeq\mathcal W_3, with \mathcal W_3 as in (1.1). Thus the height-three strong chromatic splitting formula is false in this range, whether or not an equivalence is required to preserve its specified unit summand.