Hermitian \(K\)-theory of \(A_{\infty{}}\) rings. Homotopy invariance and the hyperbolic map (Q1208679)

This paper is concerned with Hermitian \(A_ \infty\) rings, as constructed by the authors in [ibid. 519-558 (1992; see the preceding review)]. Let \(X\) be a Hermitian \(A_ \infty\) ring; this means that \(X\) is a homotopy-theoretic version of a ring with involution. The authors construct Hermitian \(K\)-theory spaces \(_ 1L(X)\) and \(_{-1}L(X)\), show that they are infinite loop spaces, and give homotopy invariance results. They also relate them to spaces of matrices, to the Hermitian \(K\)-theory of discrete rings, and to the algebraic \(K\)-theory \(K(X)\) of \(X\). In particular they construct a hyperbolic map from \(K(X)\) to \(_{\pm 1}L(X)\).