The Seifert-van Kampen theorem and generalized free products of $S$-algebras

Abstract

In a Seifert-van Kampen situation a path-connected space Z may be written as the union of two open path-connected subspaces X and Y along a common path-connected intersection W. The fundamental group of Z is isomorphic to the colimit of the diagram of fundamental groups of the three subspaces. In case the maps of fundamental groups are all injective, the fundamental group of Z is a classical free product with amalgamation, and the integral group ring of the fundamental group of Z is also a free product with amalgamation in the category of rings. In this case relations among the K-theories of the group rings have been studied. Here we describe a generalization and stablization of this algebraic fact, where there are no injectivity hypotheses on the fundamental groups and where we work in the category of S-algebras. Some of the methods we use are classical and familiar, but the passage to S-algebras blends classical and new techniques. Our most important application is a description of the algebraic K-theory of the space Z in terms of the algebraic K-theories of the other three spaces and the algebraic K-theory of spaces Nil-term.