Hermitian algebraic K-theory of simplicial rings and topological spaces
As stated by the authors in the introduction, the purpose of this paper is to extend the Hermitian algebraic K-theory of \textit{M. Karoubi} [Ann. Math., II. Ser. 112, 207-238 (1980; Zbl 0478.18008)] from rings with involution to simplicial rings with involution, just as Waldhausen extended algebraic K-theory of rings to simplicial rings. One then obtains the Hermitian algebraic K-theory of a space X with orientation \(\omega \in H^ 1(X; {\mathbb{Z}}/2)\) by applying the extension of Karoubi's theory to the simplicial group ring \({\mathbb{Z}}[\Omega_*X]\) with involution induced by \(\omega\). Given a simplicial ring \(R_*\) with involution \(\iota\) and a central element \(\epsilon\) in \(R_ 0\) such that \(\epsilon \cdot \iota (\epsilon)=1\), the authors denote the Hermitian algebraic K-theory of the triple \((R_*,\iota,\epsilon)\) by \(_{\epsilon}\tilde L(R_*,\iota)\). Here is one definition. Let \(G\tilde L_ n(R_*)\) denote the simplicial monoid of \(n\times n\) matrices over \(R_*\) which become invertible when elements are replaced by their images in \(\pi_ 0R_*\). Let \(Sym_ n^{\epsilon}(R_*)\) be the simplicial set of \(n\times n\) matrices \(A\in G\tilde L_ n(R_*)\) such that \(A^*=\epsilon A\) (where * is induced by \(\omega\) and transposition). \(G\tilde L_ n(R_*)\) acts on \(Sym_ n^{\epsilon}(R_*)\) via \((A,\lambda)\mapsto \lambda^*A\lambda\). Apply the 2-sided bar construction, obtaining \(B_ n=B(Sym_ n^{\epsilon}(R_*),G\tilde L_ n(R_*),*)\), and set \(B_{\infty}=\coprod_{n\geq 0}B_ n\). The authors show that \(B_{\infty}\) may be group-completed. This completion is \(_{\epsilon}\tilde L(R_*,\iota).\) Two important features of the theory are demonstrated. First, localized away from 2, the theory is homotopy invariant. Secondly, given \((R_*,\iota,\epsilon)\), there are natural involutions defined on \(_{\epsilon}\tilde L(R_*)\) and \(K(R_*)\) (Waldhausen's theory). These determine symmetric-antisymmetric splittings, and there are natural maps \(_{\epsilon}\tilde L(R_*)\to K(R_*)\to_{\epsilon}\tilde L(R_*)\) which preserve the splittings and induce inverse equivalences on the symmetric parts. A separate discussion shows that the antisymmetric part of \(_{\epsilon}\tilde L(R_*)\) may be identified with that of the Wall-Witt theory \(_{\epsilon}\tilde L(\pi_ 0R_*).\) The authors use their machinery to obtain information about the (simplicial) moduli spaces of topological manifolds in a fixed homotopy type.
scientific article; zbMATH DE number 3961921