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\(K_4(\mathbb{Z})\) is the trivial group. (English) Zbl 0937.19005

The algebraic \(K\)-theory of the ring of integers \(\mathbb Z\) is still quite mysterious: the groups \(K_i(\mathbb Z)\) are known in dimensions \(i\leq 3\) and some partial results have been obtained in higher dimensions. The objective of this paper is to solve the problem in dimension \(i=4\) by proving the vanishing of the group \(K_4(\mathbb Z)\). The argument is essentially based on the study of the \(K\)-theory spectrum \(K(\mathbb Z)\) of \(\mathbb Z\) and its spectrum level rank filtration \(*\simeq F_0K(\mathbb Z)\subset\cdots\subset F_kK(\mathbb Z)\subset\cdots\subset K(\mathbb Z)\), where \(F_kK(\mathbb Z)\) is the subspectrum of \(K(\mathbb Z)\) whose \(n\)-th space is built from the simplices of the \(n\)-th space of \(K(\mathbb Z)\) involving only free modules of rank \(\leq k\).
The main part of the paper is devoted to the construction a spectral sequence \(E^1_{s,t}\Longrightarrow H_{s+t}(\overline F_kK(R))\), where \(\overline F_kK(R)=F_kK(R)/F_{k-1}K(R)\) for any PID \(R\). This spectral sequence has the property that \(E^1_{s,t}=0\) when \(s<k-1\) or \(s>2k-2\). If \(R=\mathbb Z\) and \(k\geq 2\), it turns out that \(E^1_{s,0}=0\) for all \(s\) such that \(k-1\leq s\leq 2k-3\) and consequently that the spectrum \(\overline F_kK(\mathbb Z)\) is at least \((k-1)\)-connected. On the other hand, the spectrum level rank filtration induces the rank filtration spectral sequence \(E^1_{s,t}=H_{s+t}(\overline F_{s+1}K(\mathbb Z))\Rightarrow H_{s+t}(K(\mathbb Z))\). The author carefully investigates the \(E^1\)-term of that spectral sequence in dimensions \(s+t\leq 4\); in particular, the knowledge of \(E^1_{3,1}\) depends on the calculation of the first homology group \(H_1(SL_4(\mathbb Z);St_4)\) of \(SL_4(\mathbb Z)\) with coefficients in the Steinberg module \(St_4\) performed by C. Soulé [“On the \(3\)-torsion in \(K_4(\mathbb Z)\)”, Topology 39, No. 2, 259-265 (2000; Zbl 0937.19004)].
This can be combined with the computation of the \(2\)-primary algebraic \(K\)-theory of \(\mathbb Z\) provided by J. Rognes and C. Weibel [“Two-primary algebraic \(K\)-theory of rings of integers in number fields”, J. Am. Math. Soc. 13, No. 1, 1-54 (2000; Zbl 0934.19001)] and C. Weibel [“The \(2\)-torsion in the \(K\)-theory of the integers”, C. R. Acad. Sci. Paris, Sér. I 324, No. 6, 615-620 (1997; Zbl 0889.11039)] in order to show the following theorem: the spectrum homology groups of \(K(\mathbb Z)\) are \(H_i(K(\mathbb Z))\cong \mathbb Z\), \(0\), \(0\), \(\mathbb Z/2\), \(0\) for \(i=0\), \(1\), \(2\), \(3\), \(4\) respectively. Consequently, modulo finite abelian \(2\)-groups, the unit map \(S\rightarrow K(\mathbb Z)\) from the sphere spectrum to the \(K\)-theory spectrum of \(\mathbb Z\) is at least \(4\)-connected and the induced homomorphism \(\pi_4S\rightarrow \pi_4K(\mathbb Z)\cong K_4(\mathbb Z)\) is surjective: this implies that \(K_4(\mathbb Z)\) is a finite abelian \(2\)-group since \(\pi_4S=0\). It follows again from the \(2\)-primary computation that this finite abelian \(2\)-group is trivial, in other words that \(K_4(\mathbb Z)=0\).

MSC:

19D50 Computations of higher \(K\)-theory of rings
11F75 Cohomology of arithmetic groups
55P42 Stable homotopy theory, spectra
55T25 Generalized cohomology and spectral sequences in algebraic topology
20E42 Groups with a \(BN\)-pair; buildings
20J05 Homological methods in group theory
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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