Algebraic L ‐Theory, I: Foundations

Where algebraic if-theory deals with modules, Ltheory considers modules with quadratic forms. The ^-groups are of interest to topologists because they are the surgery obstruction groups, as described by Wall ([6]). Although isomorphism groups of quadratic forms have been studied before, by Witt and others, the topological applications require new algebraic methods (cf. [5], [7]). The Z-groups Ln(7r) were obtained in [6] as the solutions to a specific topological problem, leaving open the question of the algebraic framework best suited for an '^-theory'. In [1] Novikov used algebraic if-theory and the formalism of hamiltonian physics to provide such machinery, though not as coherently as might be desired. In Part I of this paper we shall use the ideas of [1] to give the foundations of i-theory over a ring with involution, A. We shall define L-groups Un(A), Vn(A), and J^,(-4) as stable isomorphism groups of ' + forms' and '+ formations' involving finitely generated (f.g.) projective, stably f.g. free, and based A -modules respectively, depending on n (mod 4) only. Part II, [2], will be devoted to a detailed study of the L-groups of the Laurent extension ring Az = A[z,z" ], with involution z K Z~. It will be shown that there exist natural direct sum decompositions