Ends of Complexes

The ends of a topological space are the directions in which it becomes noncompact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of high-dimensional compact manifolds. The book is devoted to the related theory and practice of ends, dealing with manifolds and CW complexes in topology and chain complexes in algebra. The first part develops a homotopy model of the behavior at infinity of a noncompact space. The second part studies tame ends in topology. The authors show tame ends to have a uniform structure, with a periodic shift map. They use approximate fibrations to prove that tame manifold ends are the infinite cyclic covers of compact manifolds. The third part translates these topological considerations into an appropriate algebraic context, relating tameness to homological properties and algebraic K- and L-theory. This book will appeal to researchers in topology and geometry.

The traditional applications of algebra to topology are to compact spaces. It is now necessary to also understand non-compact topological spaces, especially open manifolds. Hitherto, the relevant material has only been available in research papers (or worse, as folklore). The book makes the topology of non-compact spaces accessible to both geometric and algebraic topologists, and algebraists. Recent developments are explained, and tools for further research are provided. In short, a systematic exposition of the theory and practice of ends of manifolds and CW complexes, along with their algebraic analogues for chain complexes.