Acyclic decompositions of manifolds
This paper describes an avenue of investigation somewhat tangential to the major thrust of geometric topology during the past thirty years. Following directions set by Bing, topologists have studied generalized n- manifolds in the context of cell-like decompositions and cell-like mappings. Conditions are now well known under which the cell like-image of a manifold is itself a manifold or a manifold factor. According to the authors of this paper, ''in contrast, very little has been done with decompositions of manifolds into Z-acyclic compacts sets.'' There are some important questions about when homology n-manifolds may be the image of n-manifolds. To assist in the exploration of these questions, this paper introduces ''machinery for producing unusual and nontrivial decompositions of manifolds into Z-acyclic but not cell-like sets.'' By the use of the machinery, examples are constructed of decompositions of the n-sphere (n\(\geq 3)\) that are totally acyclic (each element is acyclic but non- cell-like). While the stated goal of the machinery is the production of examples, the paper contains excellent descriptions of general procedures for defining manifold decompositions that can be built to incorporate specified components of a compactum. These tools may have valuable applications for purposes other than the authors'. The paper also identifies a helpful criterion to determine whether a decomposition space is a generalized manifold, namely the shape property called ''nearly 1- movable''.
scientific article; zbMATH DE number 3886764