A survey of 4-manifolds through the eyes of surgery. (English) Zbl 0974.57012

Cappell, Sylvain (ed.) et al., Surveys on surgery theory. Vol. 2: Papers dedicated to C. T. C. Wall on the occasion of his 60th birthday. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 149, 387-421 (2001).
This paper is a beautiful survey (included in a collection of papers dedicated to Professor C. T. C. Wall on the occasion of his 60th birthday) concerning the topology and geometry of manifolds faced by surgery theory. This theory represents a powerful method for constructing topological (resp. smooth) manifolds and homeomorphisms (resp. diffeomorphisms) which satisfy certain homotopy conditions. In dimension 4 the topological case is very different to the smooth one. Freedman [see for example M. H. Freedman and F. S. Quinn, Topology of 4-manifolds, Princeton Math. Ser. 39 (1990; Zbl 0705.57001)] showed that the topological version of surgery theory in dimension 4 resembles the higher dimensional one rather closely. But the smooth case differs wildly from what the high dimensional theory would predict as follows from a series of papers due to Donaldson [see for example S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds (1990; Zbl 0820.57002); see of course the references of the paper under review]. The authors review the general theory of surgery and describe the most important results in dimensions 3 and 4. Then they describe precisely what the high dimensional theory predicts, and present a detailed discussion on the current state of the researches about the topological and smooth versions of surgery theory.
For the entire collection see [Zbl 0957.00062].


57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57R65 Surgery and handlebodies
57N10 Topology of general \(3\)-manifolds (MSC2010)
57R55 Differentiable structures in differential topology
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