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A decomposition theorem for \(h\)-cobordant smooth simply-connected compact \(4\)-manifolds. (English) Zbl 0843.57020
Using classical methods, we show that any two smooth, \(h\)-cobordant, compact, simply connected 4-manifolds \(M\) and \(N\) differ only in a contractible piece. More precisely, there exist decompositions \(M = M_0 \cup_\Sigma M_1\) and \(N = N_0 \cup_\Sigma N_1\), where \(M_0\) and \(N_0\) are smooth compact contractible 4-manifolds with boundary \(\Sigma\), so that \((M_1, \Sigma)\) and \((N_1,\Sigma)\) are diffeomorphic. Moreover, if \(M\) and \(N\) are closed, then \(M_1\) and \(N_1\) may be taken to be simply connected. As a corollary to the proof, we show that for any homotopy 4-sphere \(M\), there is a homology 3-sphere \(\Sigma\) and two (possibly different) smooth embeddings of \(\Sigma\) in \(\mathbb{R}^4\), such that \(M\) is obtained by gluing together along \(\Sigma\) the closures of the bounded components of \(\mathbb{R}^4 - \Sigma\).

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R80 \(h\)- and \(s\)-cobordism
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