While 2–complexes warrant study for their own complexities, the unresolved conjectures for 2–complexes described in Chapters X–XII arose from considerations of 2–dimensional spines of manifolds. This chapter investigates 2–complexes that are spines of closed orientable 3–manifolds and special 2–polyhedra that are spines of singular 3–manifolds.

3–Manifolds

This first section presents connections between combinatorial group theory and closed orientable 3–manifolds. A homotopy classification of these manifolds and an investigation of their spines are presented from the viewpoint of combinatorial squashings for them.

Representing 3–manifolds

By a 3–manifold, we mean a connected (second countable Hausdorff) space in which each point has a 3–ball neighborhood. This view of 3–manifolds as collections of overlapping 3–balls is too unstructured to contribute to their classification. To better understand 3–manifolds, especially closed orientable ones, numerous methods of representing them have been developed.

One can represent 3–manifolds as assemblages of basic bounded 3–manifolds that are glued along their boundary components in regulated manners.

Triangulations The simplest assemblage system is a triangulation, that is, a finite collection of tetrahedra whose 2–faces are identified in pairs via simplicial isomorphisms. The triangulability of closed 3–manifolds was established by E. Moise [Mo51]. This inherent structure of closed 3–manifolds contributes indirectly to their classification by offering a starting point for most other representations of 3–manifolds.

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.

This chapter presents homotopy classifications of two dimensional CW complexes and maps between them. Cases of these abstract classifications are detailed in Chapter III. Simplicial techniques are invoked in Section 1 to analyze maps of balls and spheres into 2-complexes. This analysis is applied in Section 2 to study the long exact sequence of homotopy groups for a 2-complex and to derive J. H. C. Whitehead's equivalence of the homotopy theory of 2-complexes with the purely algebraic theory of free crossed modules. Cellular chain complexes of universal coverings of 2-complexes are developed in Section 3. This equivariant world provides the foundation for the treatment in Section 4 of an abelianized version of Whitehead's equivalence, namely, the theory of algebraic 2-type of 2-complexes due to S. Mac Lane and Whitehead.

Techniques in Homotopy

In this section, we use simplicial approximations of maps between simplicial complexes to construct combinatorial approximations of maps between CW complexes, at least in dimensions one and two.

Simplicial Techniques

We view real m-space ℝm as a real vector space and we assume that the reader is familiar with the concepts of finite simplicial complexes K in ℝm and simplicial maps φ: K → M between such complexes. We don't distinguish notationally between a simplicial complex and the associated topological subspace of ℝm, and let context convey the object under consideration.