Bourgin, David G.2022-01-262022-01-26197313690049https://hdl.handle.net/10657/8606This 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.application/pdfenThis item is protected by copyright but is made available here under a claim of fair use (17 U.S.C. §107) for non-profit research and educational purposes. Users of this work assume the responsibility for determining copyright status prior to reusing, publishing, or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires express permission of the copyright holder.Thesisreformatted digital