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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 29, Issue 2

# The slice spectral sequence for the ${C}_{4}$ analog of real K-theory

Michael A. Hill
/ Michael J. Hopkins
/ Douglas C. Ravenel
Published Online: 2016-05-27 | DOI: https://doi.org/10.1515/forum-2016-0017

## Abstract

We describe the slice spectral sequence of a 32-periodic ${C}_{4}$-spectrum ${K}_{\left[2\right]}$ related to the ${C}_{4}$ norm ${\mathrm{MU}}^{\left(\left({C}_{4}\right)\right)}={N}_{{C}_{2}}^{{C}_{4}}{\mathrm{MU}}_{ℝ}$ of the real cobordism spectrum ${\mathrm{MU}}_{ℝ}$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor ${\underset{¯}{\pi }}_{*}{K}_{\left[2\right]}$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real K-theory spectrum ${K}_{ℝ}$ was first analyzed by Dugger. The ${C}_{8}$ analog of ${K}_{\left[2\right]}$ is 256-periodic and detects the Kervaire invariant classes ${\theta }_{j}$. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that ${\theta }_{j}$ does not exist for $j\ge 7$.

MSC 2010: 55Q10; 55Q91; 55P42; 55R45; 55T99

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Published Online: 2016-05-27

Published in Print: 2017-03-01

The authors were supported by DARPA Grant FA9550-07-1-0555 and NSF Grants DMS-0905160, DMS-1307896.

Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 383–447, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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