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Zbl 0533.57006
Siebenmann, Laurent
La conjecture de Poincaré topologique en dimension 4 [d'après M. H. Freedman]. (French)
[A] Sémin. Bourbaki, 34e année, Vol. 1981/82, Exp. No.588, Astérisque 92-93, 219-248 (1982).

[For the entire collection see Zbl 0489.00003.] \par The paper under review is one of the most striking achievements in the topology of manifolds. The title of the paper points out the most sensational side of it, but other results are of more general character and seem to be more interesting. They are: (1) the topological classification of closed smooth simply connected 4-manifolds; (2) the topological h-cobordism theorem for a simply connected smooth 5- dimensional h-cobordism; (3) the existence of a contractible topological 4-manifold bounded by any prescribed homology 3-dimensional sphere; (4) the topological classification of the closed simply connected topological 4-dimensional manifolds admitting a differential structure on the complement of a point (the last condition is always satisfied, as it was shown in the subsequent work of {\it F. Quinn} (see Zbl 0533.57009). \par From the technical point of view all these results are based on the previous Casson theory of flexible handles (they are called the Casson handles now) and on the theorem, which was proved by M. H. Freedman in 1981. He proved that any Casson handle is homeomorphic to the standard handle $D\sp 2\times {\bbfR}\sp 2$. The paper under review contains an interesting report on this event and a detailed exposition of the proof. Since then Freedman's paper with these results has been published [cf. {\it M. H. Freedman}, J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)].
[O.Ya.Viro]

MSC 2000:: *57N13 Topology of Euclidean 4-space, 4-manifolds
57R80 h- and s-cobordism
57R60 Homotopy spheres, Poincare conjecture

Keywords: h-cobordism in dimension five; Poincaré conjecture in dimension four; classification of closed smooth simply connected 4-manifolds; Casson handles

Citations: Zbl 0533.57009; Zbl 0489.00003; Zbl 0528.57011

Cited in: Zbl 0681.57008 Zbl 0533.57009 Zbl 0533.57008

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