# Three-dimensional manifolds

Thesis

In the post-war years, the theory of 3-dimensional manifolds has developed tremendously. On the one hand, Bing and Moise have proved that 3-manifolds can be triangulated, and that the Hauptvermutung (that any two triangulations of the same space are combinatorially equivalent) is true for 3-manifolds. On the other hand, Papakyriakopoulos has proved Dehn's Lemma, and, using ideas of Papakyriakopoulos, Whitehead has proved the Sphere Theorem. As a result of this concerted attack from two different directions, the theory of 3-manifolds has become an extremely interesting and fruitful field of study. It seems as though we are well on the way to solving the two main problems in the field:- the Poincaré Conjecture, and the classification of closed 3-manifolds. In this thesis, some theorems connected with 3-manifolds are proved. The most important theorem is the Projective Plane Theorem (6.1), in which it is proved that elements of the second homotopy group of a 3-manifold can be represented, in a certain sense, by 2-spheres or projective planes in the manifold. The Projective Plane Theorem is, perhaps, an important tool in the problem of classifying non-orientable 3-manifolds. The entire thesis depends on the Projective Plane Theorem, except for Chapters I and III. In Chapter I, the linking of n-spheres in (n+2)-space is dealt with. In Chapter Ill, non-orientable compact 3-manifolds, with finite fundamental groups are considered, with the aim of proving that there is essentially only one such 3-manifold. The reader is warned that a different definition of a 3-manifold is adopted in each chapter. This is in the interest of brevity and clarity. The author hopes that no confusion will arise. The definition appropriate in each chapter is given in the introduction to that chapter. The exact hypotheses about the 3-manifold, required for each theorem, are given just before the statement of the theorem. The follov/ing conventions are used throughout the thesis:- i) "M" denotes a 3-manifold; ii ) "X ̃" denotes some covering space of the topological space X; iii ) "G" denotes a group; iV) "O" denotes the group with only one element, or the unit element of a group which is definitely abelian, or the integer zero; v) "l" denotes the unit element of a (possibly) non-abelian group, or the integer one; vi ) "Homotopic to zero" means "homotopic to the constant map "; vii) '"The zero map" of one group into another group denotes the map which sends all elements into the trivial element. The author would like to thank Dr. E. C. Zeeman most warmly for his constant help and encouragement during the writing of this work. He could not have hoped for a better teacher. The author has found Dr. Zeeman's comments, suggestions and keen interest in his work invaluable and inspiring. The author has also had many very interesting and useful conversations with Dr. J. F. Adams, to whom he is most grateful.