Finite domination and Novikov rings

Abstract

IN order to distinguish between the combinatorial properties of finite simplicial complexes and the topology of compact polyhedra and compact manifolds it is necessary to consider infinite simplicial complexes, non-compact polyhedra, open manifolds, and algebraic Kand L-theory. The classic cases are the Milnor Hauptvermutung counterexamples of non-combinatorial homeomorphisms of compact polyhedra, the proof by Novikov of the topological invariance of the rational Pontrjagin classes, and the structure theory of Kirby and Siebenmann for high-dimensional compact topological manifolds. The open manifolds arise geometrically as tame ends: in the applications it is necessary to close them. The obstruction theory for closing tame ends of open manifolds is also the obstruction theory for deciding if a finitely dominated space is homotopy equivalent to a finite C W complex.