Point-Like, Simplicial Mappings of a 3-Sphere

Abstract

A decomposition of a topological space X is a partitioning of X into non-empty, disjoint sets called elements of the decomposition. An element of a decomposition is non-degenerate if it contains more than one point. Associated with each decomposition D of X is a topological space D*, called the hyperspace of the decomposition. A classical problem on decompositions of topological spaces is to find conditions under which D* is homeomorphic to X. Often decompositions arise from mappings: if g is a mapping of a space X onto a space Y, then D = {g-1(y) |y ∊ Y} is a decomposition of X.

Simplicial Structures and Transverse Cellularity

“…A topological theorem of R. Finney [5], combined with the Hauptvermutung in low dimensions [10], r11] plex K induced by a simplicial mapping of a subcomplex of K. (This applies in an interesting fashion to a "figure-eight decomposition" of S4.) cellular mappings of manifolds without boundary preserve combinatorial structure.…”

Concerning cellular decompositions of $3$-manifolds that yield $3$-manifolds

“…Some previous results concerning point-like decompositions of E3 and Ss for which the decomposition space is a 3-manifold are given in [1], [2], [3], [8] and [12]. Theorem 3 of [2] and the result announced in [3] are corollaries of the theorem of this paper.…”

Factoring pointlike simplicial mappings

“…This is true since the inverse image of a point is either a point or an interval. In any case g~\L) is homotopic to L by a homotopy/', t e [0,1] …”

“…In 1963, Ross Finney answered a special case of the above question [1]. He proved that if there exists a pointlike simplicial mapping from a 3-sphere onto a triangulated space T, then T is a 3-sphere.…”