Uniqueness of connected sum decompositions in dimension 4
type: Item
instance of: scholarly article
title: Uniqueness of connected sum decompositions in dimension 4
zbMATH Open document ID: 0814.57016
DOI: 10.1016/0166-8641(94)90080-9
published in: Topology and its Applications
publication date: 1995-06-18T00:00:00Z
review text: It is a well known fact that any 3-dimensional manifold decomposes uniquely as a connected sum. In 4-manifold topology this is not true as the example \(CP^ 2 \# (CP^ 2 \# -CP^ 2) = CP^ 2 \# (S^ 2 \times S^ 2)\) shows. A more subtle example is given by the homeomorphism \(^*CP^ 2 \# ^*CP^ 2 = CP^ 2 \# CP^ 2\), where \(^*CP^ 2\) is the nonsmoothable 4-manifold homotopy equivalent to \(CP^ 2\), see the book ``Topology of 4-manifolds'' (1990; Zbl 0705.57001) by \textit{M. Freedman} and \textit{F. Quinn}. To avoid such examples, the author studies decompositions with one factor fixed and inducing ``the same'' decompositions on \(\pi_ 2\). In this situation he asks whether a homeomorphism \(M \# W_ 1 = M \# W_ 2\) implies \(W_ 1 = W_ 2\)? This was proven in the above book for good fundamental groups under the assumption that \(M\) is not spherically characteristic. Assuming again that the fundamental group is good, the author extends this result to other cases, most notably to the case when \(M\) is a spin manifold. As a nice corollary he proves that the \(^*\)-construction of Freedman and Quinn (loc. cit.) is well defined.
reviewed by: Peter Teichner
Mathematics Subject Classification ID: 57N13
zbMATH DE Number: 567712
zbMATH Keywords: 4-manifolds
zbMATH Keywords: connected sum
zbMATH Keywords: good fundamental groups
zbMATH Keywords: spin manifold
zbMATH Keywords: \(^*\)-construction
author: Richard A. Stong
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)90080-9
OpenAlex ID: W1979311570
cites work: L-theory and the Arf invariant
cites work: On the homotopy theory of simply connected four manifolds
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cites work: Four-dimensional topology: an introduction
cites work: Uniqueness of \(\pi_ 1\)-negligible embeddings in 4-manifolds: A correction to theorem 10.5 of Freedman and Quinn
cites work: Q5687301
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description: scientific article; zbMATH DE number 567712