The Algebraic Theory of Surgery I. Foundations

algebra. An n-dimensional algebraic Poincare complex over a ring A with an involution -: A -+ A; a r+ a is an A-module chain complex G with an n-dimensional Poincare duality H*(G) = Hn_*(G). We shall use n-dimensional algebraic Poincare complexes to define two sequences of covariant functors Ln {Ln}: (rings with involution) -+ (abelian groups) (n E Z) . such that LO(A) {respectively Lo(A)} is the Witt group of non-singular symmetric {quadratic} forms over A. The quadratic L-groups Ln(A) will turn out to be the surgery obstruction groups of Wall [25], with a 4-periodicity