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Generalized surgeries of three-dimensional manifolds and representations of homology spheres. (English) Zbl 0634.57006

The author gives some necessary and sufficient conditions for a 3- manifold to be a homology sphere and for two 3-manifold to have isomorphic homology groups and linking forms. These conditions are obtained in terms of a new concept of generalized surgery and are restated in classical terms of Dehn surgeries. The main novelty of the paper is this concept of surgery.
Let us fix a triple \({\mathcal V}=(V,V_+,V_-)\), where \({\mathcal V}\) is a closed n-manifold and \(V_+,V_-\) are submanifolds of codimension, such that \(V=V_+\cup V_-\) and \(V_+\cap V_-=\partial V_+\cap \partial V_-\). Let \(M_ 1\) be a closed n-manifold and \(h: V_-\to M_ 1\) be an embedding. Then \(M_ 2=(M_ 1-Int h(V_-))\cup_{h| \partial V_+}V_+\) is defined to be the result of a \({\mathcal V}\)-surgery on \(M_ 1\). In addition, the roles of \(V_+,V_-\) can be exchanged. Two examples of this notion are studied in the paper. In the first one \((V,V_+,V_-)\) is a genus 2 Heegaard splitting of the Whitehead manifold, in the second one \((V,V_+,V_-)\) is a genus 3 Heegaard splitting of S \(1\times S\) \(1\times S\) 1.
Reviewer: N.Ivanov

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57R65 Surgery and handlebodies
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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