The slice spectral sequence for the C₄ analog of real K-theory

2015/02/26 by Hill, Michael A., Hopkins, Michael J., Ravenel, Douglas C. · 2 citations
#55P42 #55Q10 (primary) #55R45 #55T99 (secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #and 55Q91

paper · doi:10.48550/arxiv.1502.07611

Abstract

We describe the slice spectral sequence of a 32-periodic C₄-spectrum K_[2] related to the C₄ norm N_C₂^C₄MU_{\bf R} of the real cobordism spectrum MU_{\bf R}. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor \underlineπ_*K_[2], complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real K-theory spectrum K_{\bf R} was first analyzed by Dugger. The C₈ analog of K_[2] is 256-periodic and detects the Kervaire invariant classes θ_j in the stable homotopy groups of spheres. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that θ_j does not exist for j≥ 7.

Cited by

• Periodicity and finite complexity in higher real K-theories
• On higher real K-theories and finite spectra

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