A decomposition theorem for \(h\)-cobordant smooth simply-connected compact \(4\)-manifolds
type: Item
instance of: scholarly article
title: A decomposition theorem for \(h\)-cobordant smooth simply-connected compact \(4\)-manifolds
zbMATH Open document ID: 0843.57020
published in: Inventiones Mathematicae
publication date: 1996-08-18T00:00:00Z
full work available at URL: https://eudml.org/doc/144350
review text: 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 \(\mathbb{R}^4 - \Sigma\).
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R80
zbMATH DE Number: 856594
zbMATH Keywords: \(h\)-cobordant
zbMATH Keywords: 4-manifold
zbMATH Keywords: homotopy 4-spheres
author: Cynthia L. Curtis
author: Michael Freedman
author: Wu-Chung Hsiang
author: Richard A. Stong
reviewed by: Cynthia L. Curtis
MaRDI profile type: Publication
OpenAlex ID: W4236458155
DOI: 10.1007/S002220050031
description: scientific article; zbMATH DE number 856594