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The disk theorem for four-dimensional manifolds. (English) Zbl 0577.57003

Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 647-663 (1984).
[For the entire collection see Zbl 0553.00001.]
At the heart of manifold topology (in any dimension) is the understanding of mappings of the 2-disc \(D^ 2\). Four dimensional manifolds are particularly sensitive to immersions of \(D^ 2\)- any question concerning 4-manifolds can ultimately be reduced to a question concerning such immersions. In this important paper, the author presents his 4- dimensional 2-disc imbedding theorem. The simply connected version was the key of his earlier work on the topological 4-dimensional PoincarĂ© conjecture [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)]. This disc theorem has as useful corollaries appropriate s-cobordism and non- simply connected surgery theorems. The fundamental groups for which the disc theorem is proven can be described as: (1) containing all finite groups and the integers, and (2) being closed under: subgroup, quotient group, extension, and (infinite) nested union.
This well-written paper is not the end of the theory. There remain exciting open problems (does the disc embedding theorem hold for free fundamental groups?) with even more exciting consequences.
Reviewer: R.Stern

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)