Generalised Vietoris-Begle theorems

This dissertation investigates Vietoris-Begle type theorems for sheaf theoretic homology, and explores the possibility of establishing a Vietoris-Begle type theorem for a more general functor which is constructed on a category of inverse systems. The first part (Chapter 2) of the dissertation is devoted to a brief study of two basic cone constructions, and the almost p-solid condition, both introduced in recent papers of D. G. Bourgin [4], [5], It is demonstrated that the almost p-solid condition guarantees that certain topological properties are preserved under cone constructions. In the second section of this chapter it is proved that the two cone spaces are homeomorphic. Finally, a so-called generalized mapping cylinder is introduced and it is shown that the cone spaces are homeomorphic to a subspace of this mapping cylinder. In Chapter 3, Vietoris-Begle type theorems and their inverses are constructed for locally compact spaces using sheaf theoretic homology. Applications are given to Wilder's monotone theorem [11], and to a generalization of the Vietoris-Begle theorem to triple spaces given by Bialynicki-Birula [2]. In the final chapter, a contravariant functor H is constructed on a category of inverse systems. An underlying category U of topological pairs is proved to be admissible for a cohomology theory in the sense of Eilenberg and Steenrod [10]. It is shown that Vietoris-Begle type maps are admissible for the category U. However, it does not appear that a Vietoris-Begle theorem of any generality can be exhibited for the original functor H. For this reason the conditions on the construction of H are relaxed and Vietoris-Begle type theorems are proved.