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Ends of maps. III: Dimensions 4 and 5. (English) Zbl 0533.57009
[A preliminary version was given in Prepr. Ser., Aarhus Univ. 1982/83, No.4, 33 p. (1982).]
In this paper the author proves 5-dimensional versions of his thin h- cobordism and end theorems proven in dimensions \(\geq 6\) in Part I of this paper [see the preceding review]. These theorems directly imply several well sought after facts about 4- and 5-manifolds. For example: (1) The stability map \(TOP(4)/O(4)\to TOP/O\) is 3-connected. This implies that every topological 4-manifold is smooth in the complement of a point. (2) Topological 5-manifolds have handlebody structures, relative to arbitrary submanifolds of their boundary. Thus, all topological manifolds have handlebody structures except nonsmoothable 4-manifolds. (3) Map transversality holds in all dimensions. Submanifold transversality holds except in ambient dimension 4 and when the dimension of intersection is \(\geq 1\). (4) A cell-like map between 4-manifolds can be approximated by homeomorphisms. (5) ANR homology manifolds of dimension 4 have resolutions. (6) The 4-dimensional annulus conjecture.
The key to the proof of the thin h-cobordism and end theorems is a disc deployment lemma which disjointly embeds an infinite number of discs with \(\epsilon\)-control. Essentially, this is all that is missing from the author’s high dimensional proof. The new ingredient in dimension 4 is M. Freedman’s disc embedding theorem which allows to embed discs in Casson handles.
Reviewer: R.Stern

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57N75 General position and transversality
57N60 Cellularity in topological manifolds
57N35 Embeddings and immersions in topological manifolds
57N30 Engulfing in topological manifolds
57R80 \(h\)- and \(s\)-cobordism
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