Document Zbl 1412.19001 - zbMATH Open

Let \(\mathbb{S}\) be the sphere spectrum. The authors compute the algebraic \(K\)-theory of \(\mathbb{S}\), \(K(\mathbb{S})\). As the free part is well understood and previous calculatins have been performed by Rognes at regular primes, this paper calculates \(\pi_{*}K(\mathbb{S})_{p}^{\land}\) for irregular primes \(p\). Based on trace methods, the first result is that for an odd prime \(p\) there is a noncanonical split short exat sequence \[ 0\to \pi_{*}K(\mathbb{S})_{p}^{\land}\to \pi_{*}TC(\mathbb{S})_{p}^{\land} \oplus \pi_{*}K(\mathbb{Z})_{p}^{\land}\to \pi_{*}TC(\mathbb{Z})_{p}^{\land}\to 0, \] where \(TC\) denotes topological cyclic homolgy.
The authors prove the following Theorem 1.2. Let \(p\) be an odd prime. The \(p\)-torsion in \(\pi_{*}K(\mathbb{S})\) admits canonical isomorphisms: \[ \begin{aligned} \mathrm{tor}_{p}(\pi_{*}K(\mathbb{S}))&\cong \mathrm{tor}_{p}(\pi_{*}c \oplus \pi_{*-1}c\oplus \pi_{*}\overline{Cu}\oplus \pi_{*}K(\mathbb{Z})) \\ &\cong \mathrm{tor}_{p}(\pi_{*}\mathbb{S} \oplus \pi_{*-1}c\oplus \pi_{*}\overline{Cu}\oplus \pi_{*}K^{red}(\mathbb{Z})), \end{aligned} \] where \(c\) denotes the additive \(p\)-complete cokernel of the \(J\) spectrum, and \(Cu\) appears in the canonical splitting \(TC(\mathbb{S})\simeq \mathbb{S}\vee Cu\), where \(Cu\) is the homotopy cofiber of the unit map \(\mathbb{S}\to TC(\mathbb{S})\) and the above short exact sequence.

MSC:

19D10	Algebraic \(K\)-theory of spaces
55Q10	Stable homotopy groups

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