Curtis, C. L.; Freedman, M. H.; Hsiang, W. C.; Stong, R. A decomposition theorem for \(h\)-cobordant smooth simply-connected compact \(4\)-manifolds. (English) Zbl 0843.57020 Invent. Math. 123, No. 2, 343-348 (1996). 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 \(\mathbb{R}^4 - \Sigma\). Reviewer: C.L.Curtis (Princeton) Cited in 12 ReviewsCited in 26 Documents MSC: 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) 57R80 \(h\)- and \(s\)-cobordism Keywords:\(h\)-cobordant; 4-manifold; homotopy 4-spheres PDFBibTeX XMLCite \textit{C. L. Curtis} et al., Invent. Math. 123, No. 2, 343--348 (1996; Zbl 0843.57020) Full Text: DOI EuDML