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

Paul Goerss¹ ; Hans-Werner Henn² ; Mark Mahowald³ ; Charles Rezk⁴

¹ Department of Mathematics, Northwestern University, Evanston, IL 60208, United States
² Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 67084 Strasbourg, France
³ Department of Mathematics, Northwestern University, Evanston IL 60208, United States
⁴ Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States

Annals of mathematics, Tome 162 (2005) no. 2, pp. 777-822.

Résumé

We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava $K$-theory $K(2)$. At the prime $3$, we write the spectrum $L_{K(2)}S^0$ as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form $E_2^{hF}$ where $F$ is a finite subgroup of the Morava stabilizer group and $E_2$ is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case $n=2$ at $p=3$ represents the edge of our current knowledge: $n=1$ is classical and at $n=2$, the prime $3$ is the largest prime where the Morava stabilizer group has a $p$-torsion subgroup, so that the homotopy theory is not entirely algebraic.

Détail

DOI : 10.4007/annals.2005.162.777