Asymmetric four-dimensional manifolds
type: Item
instance of: scholarly article
title: Asymmetric four-dimensional manifolds
zbMATH Open document ID: 0669.57022
DOI: 10.1215/S0012-7094-86-05341-X
author: Pierre Vogel
author: Slawomir Kwasik
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The main result of this paper is a proof of the existence of simply connected closed manifolds which admit no locally smooth involutions. The principal example is the 4-manifold \(E_ 8\). (This is the closed simply connected 4-manifold with intersection form \(E_ 8\) constructed by \textit{M. H. Freedman} [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)].) The nonexistence of an involution depends on the fact that the Kirby- Siebenmann obstruction to smoothability is nontrivial for \(E_ 8\). In fact the authors' proof shows that any closed 4-manifold admitting a locally smooth involution must have vanishing Kirby-Siebenmann obstruction. The authors construct an infinite number of examples by considering the connected sum of \(E_ 8\) with copies of \(S^ 2\times S^ 2\). The arguments work as well if \(E_ 8\) is replaced with the Chern manifold, the nonsmoothable 4-manifold which is homotopy equivalent to \({\mathbb{C}}P^ 2\). The existence of this manifold is also due to Freedman.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R10
zbMATH DE Number: 4096402
zbMATH Keywords: simply connected closed manifolds which admit no locally smooth involutions
zbMATH Keywords: simply connected 4-manifold with intersection form \(E_ 8\)
zbMATH Keywords: Kirby-Siebenmann obstruction to smoothability
zbMATH Keywords: Chern manifold
MaRDI profile type: Publication
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full work available at URL: s0012-7094-86-05341-x
OpenAlex ID: W2073019443
description: scientific article; zbMATH DE number 4096402