##
**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.

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

### MSC:

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 |