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The intimate connections among decomposition theory, embedding theory, and manifold structure theory. (English) Zbl 0631.57011

Geometric topology and shape theory, Proc. Conf., Dubrovnik/Yugosl. 1986, Lect. Notes Math. 1283, 43-47 (1987).
[For the entire collection see Zbl 0619.00017.]
This survey details examples that show an intimate connection among decomposition theory, embedding theory, and manifold structure theory. The idea of such a connection was first articulated by J. W. Cannon [Lect. Notes Math. 438, 66-118 (1975; Zbl 0306.57004)]. Cannon’s observation was based on work begun in the early fifties by R. H. Bing and has proved to be prophetic for the work that was to follow.
This paper gives a very convincing argument concerning ten areas of research in a period of over thirty years. It includes discussions of nonstandard involutions of a 3-sphere, the generalized Schoenflies theorem, non-manifold factors of Euclidean space, homology spheres bound contractible manifolds, the solution to the annulus conjecture, local contractability of manifold homeomorphism groups, near homeomorphisms between manifolds, approximations of codimension at least three compacta by “nice” embeddings, necessary and sufficient conditions for codimension one manifolds to be locally flat, approximations of codimension one manifolds by locally flat embeddings, characterization of shrinkable decompositions of manifolds, characterization of topological manifolds, and the 4-dimensional Poincaré conjecture.
This survey makes a good case that decomposition theory has an important place in geometric topology in much the same way as dimension theory or algebraic topology.
Reviewer: D.G.Wright

MSC:

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57P99 Generalized manifolds
54B15 Quotient spaces, decompositions in general topology
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M30 Wild embeddings
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57N45 Flatness and tameness of topological manifolds
57N60 Cellularity in topological manifolds