Editors' Preface

Summary

It is well known that techniques developed for manifolds of higher dimensions don't suffice to treat open problems in dimensions three and four. The latter are inextricably tied to questions about the (simple-) homotopy type of 2-skeleta or -spines of these low-dimensional manifolds and, hence, to presentations of groups.

Basic work on two-dimensional homotopy dates back to K. Reidemeister and J.H.C. Whitehead. For instance, Whitehead gave an algebraic description of the homotopy type of 2-complexes. But, until the early 70's, one didn't have examples of 2-complexes with different homotopy type but equal fundamental groups and Euler characteristic. Since then considerable advances have been made, yielding, in particular, remarkable partial results on famous open problems like Whitehead's question, whether subcomplexes of aspherical 2-complexes are always aspherical themselves. The authors of this book have contributed to this development.

Because of its relations to decision problems in combinatorial group theory, two-dimensional homotopy probably will never take the shape of a complete theory. However, the occurrence of certain notions (e.g., the Reidemeister-Peiffer identities of presentations) in different questions is far from being accidental. The time has come to collect the present knowledge in order to stimulate further research.

This book contains the elements of both a textbook and a research monograph, and, hence, addresses students as well as specialists. Parts of the book have already been used to substantiate courses with concrete geometric and/or algebraic material. A student reader should know already some (algebraic) topology and algebra. We start with two introductory chapters on low-dimensional complexes. These are followed by chapters on prominent techniques including their applications to manifolds.