A resolution of the \(K(2)\)-local sphere at the prime 3.

*(English)*Zbl 1108.55009Since the 1970’s and the successes of Miller, Ravenel, and Wilson in organizing the periodic phenomena in the stable homotopy groups of spheres, the emergence of a clearer qualitative picture has turned on the chromatic viewpoint established by Devinatz, Hopkins, and Smith. The chief engine behind much of this progress has been the algebraic geometry of formal groups along with its manifestation after localization. The calculations of Shimomoru and his collaborators provide the grist for this paper—data that call out for explanation within this framework.

The main result is a sequence of maps between spectra \[ \begin{align*}{ L_{K(2)}S^0 \to E_2^{hG_{24}} &\to \Sigma^8 E_2^{hSD_{16}} \vee E_2^{hG_{24}} \to \Sigma^8 E_2^{hSD_{16}} \vee \Sigma^{40} E_2^{hSD_{16}} \cr &\to \Sigma^{40} E_2^{hSD_{16}} \vee \Sigma^{48} E_2^{hSD_{16}} \to \Sigma^{48} E_2^{hSD_{16}}\cr}\end{align*} \] for which any composite of two consecutive maps is zero and all possible Toda brackets are zero modulo indeterminacy. The Lubin-Tate spectra \(E_n\) are defined within the \(K(n)\)-local category to have a certain formal group law realized as the universal deformation of the Honda formal group law and homotopy groups a power series ring over Witt vectors. In the \(K(n)\)-local category \(E_n\) is a wedge of spectra of the form \(L_{K(n)}E(n)\) with \(E(n)\) the Johnson-Wilson homology theories. In the case of \(E_2\) over the prime 3, there are actions of certain groups: the group \(G_{24}\) is an extension of a maximal finite subgroup \(G_{12}\) of the Morava stabilizer group \(S_2\) and the Galois group of the \(\mathbb F_9\) over \(\mathbb F_3\). The other subgroup \(SD_{16}\subset S_2\) is expressible as a semidirect product of \(\mathbb F_9^\times\) and \(\mathbb Z/2\), the semidihedral group of order 16.

Although this paper is highly technical (as all progress in the computation of the stable homotopy groups of spheres must be), the authors have provided ample discussion of their framework and its success establishes the importance of the viewpoint for homotopy theory.

The main result is a sequence of maps between spectra \[ \begin{align*}{ L_{K(2)}S^0 \to E_2^{hG_{24}} &\to \Sigma^8 E_2^{hSD_{16}} \vee E_2^{hG_{24}} \to \Sigma^8 E_2^{hSD_{16}} \vee \Sigma^{40} E_2^{hSD_{16}} \cr &\to \Sigma^{40} E_2^{hSD_{16}} \vee \Sigma^{48} E_2^{hSD_{16}} \to \Sigma^{48} E_2^{hSD_{16}}\cr}\end{align*} \] for which any composite of two consecutive maps is zero and all possible Toda brackets are zero modulo indeterminacy. The Lubin-Tate spectra \(E_n\) are defined within the \(K(n)\)-local category to have a certain formal group law realized as the universal deformation of the Honda formal group law and homotopy groups a power series ring over Witt vectors. In the \(K(n)\)-local category \(E_n\) is a wedge of spectra of the form \(L_{K(n)}E(n)\) with \(E(n)\) the Johnson-Wilson homology theories. In the case of \(E_2\) over the prime 3, there are actions of certain groups: the group \(G_{24}\) is an extension of a maximal finite subgroup \(G_{12}\) of the Morava stabilizer group \(S_2\) and the Galois group of the \(\mathbb F_9\) over \(\mathbb F_3\). The other subgroup \(SD_{16}\subset S_2\) is expressible as a semidirect product of \(\mathbb F_9^\times\) and \(\mathbb Z/2\), the semidihedral group of order 16.

Although this paper is highly technical (as all progress in the computation of the stable homotopy groups of spheres must be), the authors have provided ample discussion of their framework and its success establishes the importance of the viewpoint for homotopy theory.

Reviewer: John McCleary (Poughkeepsie)