Introduction

The subject of this chapter is the rather audacious conjecture put forward in 1964 by E. C. Zeeman.

Zeeman's Conjecture (Z)If P2is a contractible 2-dimensional polyhedron, then P2is 1-collapsible, that is, P2 × I collapses to a point.

As already pointed out in Chapter I, §4.2, (Z) implies both the Poincaré conjecture (P) and the Andrews-Curtis conjecture (AC). It is an affirmation of the subtlety of low-dimensional topology that these old basic conjectures are still unsolved, despite strenuous efforts of generations of topologists. The attempts to solve (Z) have led mathematicians to discover novel ideas and powerful methods in low-dimensional topology, and to a deeper understanding of the strange and mysterious world of 2-dimensional complexes.

Although there are many candidates for counterexamples, (Z) has not been refuted (if it is false!) because, for the present, we have no methods of detecting non-collapsibility for contractible 3-dimensional polyhedra of the form P2 × I. As a result, the main achievements in investigation of (Z) consist in

1. proving it for different special types of P2;

2. proving of weakened and disproving of strengthened versions of (Z).

The first contributions to (1) were made by P. Dierker, W. B. R. Lickorish, D. Gillman [Di68, Li70, Gi86] and may be called collapsing by adding a cell.