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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.
Paul Goerss 1 ; Hans-Werner Henn 2 ; Mark Mahowald 3 ; Charles Rezk 4
@article{10_4007_annals_2005_162_777,
author = {Paul Goerss and Hans-Werner Henn and Mark Mahowald and Charles Rezk},
title = {A resolution of the {K(2)-local} sphere at the prime 3},
journal = {Annals of mathematics},
pages = {777--822},
publisher = {mathdoc},
volume = {162},
number = {2},
year = {2005},
doi = {10.4007/annals.2005.162.777},
mrnumber = {2183282},
zbl = {1108.55009},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2005.162.777/}
}
TY - JOUR AU - Paul Goerss AU - Hans-Werner Henn AU - Mark Mahowald AU - Charles Rezk TI - A resolution of the K(2)-local sphere at the prime 3 JO - Annals of mathematics PY - 2005 SP - 777 EP - 822 VL - 162 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2005.162.777/ DO - 10.4007/annals.2005.162.777 LA - en ID - 10_4007_annals_2005_162_777 ER -
%0 Journal Article %A Paul Goerss %A Hans-Werner Henn %A Mark Mahowald %A Charles Rezk %T A resolution of the K(2)-local sphere at the prime 3 %J Annals of mathematics %D 2005 %P 777-822 %V 162 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2005.162.777/ %R 10.4007/annals.2005.162.777 %G en %F 10_4007_annals_2005_162_777
Paul Goerss; Hans-Werner Henn; Mark Mahowald; Charles Rezk. A resolution of the K(2)-local sphere at the prime 3. Annals of mathematics, Tome 162 (2005) no. 2, pp. 777-822. doi: 10.4007/annals.2005.162.777
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