We study two "disjoint disks properties" in dimension 3 due to H. W. Lambert and R. B. Sher (Pacific. J. Math. 24 (1968) 511-518), the Dehn lemma property (DLP) and the map separation property (MSP). Theorem 1. Let G be a cell-like closed 0-dimensional upper semicontinuous decomposition of a 3-manifold M (possibly with boundary) with N(,G)(L-HOOK) int M. Then the following statements are equivalent: (i) M/G has the DLP; (ii) M/G has the MSP; (iii) M/G is a 3-manifold. Theorem 2. Let C be the class of all compact generalized 3-manifolds X with dim S(X) (LESSTHEQ) 0 and let C(,0)(L-HOOK) C be the subclass of all X(ELEM)C with S(X) (L-HOOK) {pt} and X (TURNEQ) S('3). Then the following statements are equivalent: (i) The Poincare conjecture in dimension three is true; (ii) If X(ELEM)C has the DLP or the MSP then S(X) = (SLASHCIRC); (iii) If X(ELEM)C(,0) has the DLP or the MSP then S(X) = (SLASHCIRC). / We also study neighborhoods of peripherally 1-acyclic compacta in nonorientable 3-manifolds. We prove a finiteness and a neighborhood theorem for such compacta and as an application extend a result of J. L. Bryant and R. C. Lacher concerning resolutions of almost (,2)-acyclic images of orientable 3-manifolds (Math. Proc. Camb. Phil. Soc. 88 (1980) 311-320), to nonorientable 3-manifolds. Theorem 3. Let f be a closed, monotone mapping from a 3-manifold M onto a locally simply connected (,2)-homology 3-manifold X. Suppose that there is a 0-dimensional set Z(L-HOOK)X such that H('1)(f('-1)(x); (,2)) = 0 for all x (ELEM) X - Z. Then the set C = {x(ELEM)X(VBAR)f('-1)(x) is not cell-like} is locally finite in X. Moreover X has a resolution. / Included is an investigation of the basic properties of generalized 3-manifolds with boundary, a topics on which little study has been done so far, as well as some results on regular neighborhoods of compacta in 3-manifolds with applications to homotopic PL embeddings of compact polyhedra into 3-manifolds. / Source: Dissertation Abstracts International, Volume: 44-02, Section: B, page: 0519. / Thesis (Ph.D.)--The Florida State University, 1983.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_75066 |
| Contributors | REPOVS, DUSAN., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 125 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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