Wiginton, C. Lamar2022-06-222022-06-22196813678515https://hdl.handle.net/10657/9726Let f be a monotone simplicial function from a triangulated combinatorial n-sphere. S[raised n] onto a triangulated, combinatorial n-manifold M[raised n]. It is shown that f is point-like if and only if f[raised -1](v) is algebraically [n/2] - connected for each vertex v of M[raised n], provided that n=3. The proof is accomplished along lines which it is hoped will lead to a proof for other dimensions, perhaps for all n, in at least a modified version of the statement. Several related conjectures are investigated by showing that some of the. lemmas used to prove the main theorem are true in all dimensions, or at least in all but two. The particular difficulties encountered by the author in trying to prove these conjectures are explained.application/pdfenThis item is protected by copyright but is made available here under a claim of fair use (17 U.S.C. Section 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