---
- 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

## Meta

- **URL**: <https://www.sciencedirect.com/science/article/pii/S0040938399000075>
- **URI**: <http://zotero.org/users/1049732/items/4F822CE7>
- **Local File**: [Rognes - 2000 - K4(Z) is the trivial group.pdf](file:///home/zack/Zotero/storage/FVRT64YC/Rognes%20-%202000%20-%20K4(Z)%20is%20the%20trivial%20group.pdf)
- **Open in Zotero**: [Zotero](zotero://select/library/items/4F822CE7)

## 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).

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