- CiteKey: “Rog00” - Type: journalArticle - Title: “K4(Z) is the trivial group,” - Author: “Rognes, John;” - Year: 2000 - DOI: 10.1016/S0040-9383(99)00007-5 - Collections: “Syllabus; Talbot 2022,” - Keywords: “Algebraic -theory of the integers”; “Component filtration”; “Poset filtration”; “Spectrum level rank filtration”; “Stable building”
K4(Z) is the trivial group
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- URL: https://www.sciencedirect.com/science/article/pii/S0040938399000075
- URI: http://zotero.org/users/1049732/items/4F822CE7
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Abstract
We prove that the fourth algebraic K-group of the integers is the trivial group, i.e., that K4(Z)=0. The argument uses rank-, poset- and component filtrations of the algebraic K-theory spectrum K(Z) from Rognes (Topology 31 (1992) 813–845; K-Theory 7 (1993) 175–200), and a group homology computation of H1(SL4(Z);St4) from Soulé, to compute the odd primary spectrum homology of K(Z) in degrees ⩽4. This shows that the odd torsion in K4(Z) is trivial. The 2-torsion in K4(Z) was shown to be trivial in Rognes and Weibel (J. Amer. Math. Soc., to appear).