A SURVEY OF WALL'S FINITENESS OBSTRUCTION

Abstract

Wall's finiteness obstruction is an algebraic K-theory invariant which decides if a finitely dominated space is homotopy equivalent to a finite CW complex. The invariant was originally formulated in the context of surgery on CW complexes, generalizing Swan's application of algebraic K-theory to the study of free actions of finite groups on spheres. In the context of surgery on manifolds, the invariant first arose as the Siebenmann obstruction to closing a tame end of a non-compact manifold. The object of this survey is to describe the Wall finiteness obstruction and some of its many applications to the surgery classification of manifolds.