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Round handles, logarithmic transforms and smooth 4-manifolds. (English) Zbl 1297.57064

An \(m\)-dimensional round \(k\)-handle is \(S^{1}\) times an \((m-1)\)-dimensional \(k\)-handle, that is an \(S^{1} \times D^{k} \times D^{m-k-1}\) attached along \(S^{1} \times \partial D^{k} \times D^{m-k-1}\). A logarithmic transform of a \(4\)-manifold is analogous to a Dehn surgery on a 3-manifold except that is is performed using an embedded \(2\)-torus with trivial normal bundle in the \(4\)-manifold. R. J. Stern posed [in: D. A. Ellwood (ed.) et al., Floer homology, Gauge theory, and low-dimensional topology. Proceedings of the Clay Mathematics Institute 2004 summer school, Budapest, Hungary, June 5–26, 2004. Providence, RI: American Mathematical Society (AMS). Cambridge, MA: Clay Mathematics Institute. Clay Mathematics Proceedings 5, 225–239 (2006; Zbl 1106.57024)] the following two questions 1. Are any two arbitrary closed smooth oriented simply connected \(4\)-manifolds \(X\) and \(X'\) in the same homeomorphism class related by a sequence of logarithmic transforms along tori? and 2. Is there a cobordism between \(X\) and \(X'\) which is composed of round \(2\)-handles only? This paper answers these two questions in the affirmative. Further the paper shows that if the four manifold \(X_{m+1}\) is obtained from \(X_{m}\) by a logarithmic transform then \(X_{m+1}\) and \(X_{m}\) become diffeomorphic after one stabilization by taking a connected sum with \(S^{2} \times S^{2}\). The proofs rely on geometric arguments.

MSC:

57R50 Differential topological aspects of diffeomorphisms
57R65 Surgery and handlebodies
57R90 Other types of cobordism

Citations:

Zbl 1106.57024
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