K4(Z) is the trivial group

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).