The upper triangular group and operations in algebraic K-theory
Let \(bu\) and \(bo\) denote the stable homotopy spectra representing 2-adically completed unitary and orthogonal connective \(K\)-theory respectively. The author considers the group \(\text{Aut}^0 (bu\wedge bo)\), under composition, of homotopy classes of left \(bu\)-module homotopy equivalences of \(bu\wedge bo\) which induces the identity map on \(H_*(bu\wedge bo\); \(\mathbb{Z}/2)\). Let \(U_\infty \mathbb{Z}_2\) denote the group of infinite, invertible upper triangular matrices with entries in 2-adic integers. The main result of the present paper is: There is an isomorphism of the form \(\psi:\text{Aut}^0 (bu \wedge bo)\to U_\infty \mathbb{Z}_2\) (Theorem 1.2). The proof uses essentially the author's prior result concerning a stable decomposition of \(\Omega^2S^3\) such that \(\Omega^2S^3 \simeq\bigvee_{k\geq 1} F_k/F_{k-1}\) where \(F_k\)'s denote an increasing filtration of \(\Omega^2S^3\) with \(F_1=S^1\). Using this splitting the author first gives the following preparatory theorem: There is a 2-local homotopy equivalence of left \(bu\)-module spectra of the form \(\widehat L:\bigvee_{k \geq 0}bu\wedge (F_{4k}/F_{4k-1})\to bu\wedge bo\) (Theorem 2.3, (ii)). Here \(F_0= *\) is replaced by \(S^0\) and \(F_{-1}=*\). The proof of this theorem requires a fair amount of homological technique applied to module spectra in order to construct the map \(\widehat L\). The key to the author's approach is using the graded homotopy-ring structure on \(\bigvee_{k\geq 1} F_k/F_{k-1}\) inherited from \(\Omega^2S^3\). Indeed this induces that on \(\bigvee_{k\geq 0}bu \wedge(F_{4k}/F_{4k-1})\) appropriate to construct a bijection from \(U_\infty (\mathbb{Z}_2)\) to \(\text{Aut}^0 (\bigvee_{k\geq 0} bu\wedge (F_{4k}/F_{4k-1}))\). Thus the discussion of Sections 2 and 3 yields the isomorphism \(\psi\). As an application, in Section 4 the author gives a family of operations on algebraic \(K\)-theory of \(F\)-schemes for an algebraically closed field \(F\) of characteristic different from 2. The last section deals with the map \(1\wedge\psi^3\) of \(bu\wedge bo\) into itself where \(\psi^3\) denotes the Adams operation. The author tries here to describe it in a matrix form and this section is closed with the following statement in the footnote: The author has recently shown that \(1\wedge\psi^3\) becomes conjugate in \(U_\infty \mathbb{Z}_2\) to a certain explicit matrix.
scientific article; zbMATH DE number 1838831
scientific article