L-theory

Definition

Let $R$ be a ring with involution $a \mapsto \bar a$ and let $\varepsilon \in \set{1, -1}$. For a finitely generated free right $R$-module $M$, let $M^* = \Hom_R(M, R)$, made a right module by the involution. For $\psi \colon M \to M^*$ let $\psi^* \colon M \to M^*$ be its dual, using $M^{**} \cong M$. An $\varepsilon$-quadratic form is a pair $(M, \psi)$ with $\psi$ taken modulo maps of the form $\chi - \varepsilon \chi^*$. It is nonsingular if $\lambda = \psi + \varepsilon \psi^* \colon M \to M^*$ is an isomorphism. The hyperbolic form $H_\varepsilon(N)$ on $N \oplus N^*$ is given by $\psi(x, f)(y, g) = f(y)$. Let $k \ge 0$. The group $L_{2k}(R)$ is the set of nonsingular $(-1)^k$-quadratic forms on free modules modulo the relation below, with orthogonal sum. $(M, \psi) \sim (M', \psi')$ if $(M, \psi) \oplus H_{(-1)^k}(N) \cong (M', \psi') \oplus H_{(-1)^k}(N')$ for some free modules $N, N'$. The odd groups $L_{2k+1}(R)$ are defined in the same spirit from nonsingular $(-1)^k$-quadratic formations, or from automorphisms of hyperbolic forms. The groups are $4$-periodic: $L_{n+4}(R) = L_n(R)$.