Decomposition of \(S^ 4\) as a twisted double of a certain manifold
type: Item
instance of: scholarly article
title: Decomposition of \(S^ 4\) as a twisted double of a certain manifold
zbMATH Open document ID: 0888.57022
author: Yuichi Yamada
published in: Tokyo Journal of Mathematics
publication date: 1998-05-25T00:00:00Z
review text: A closed oriented manifold \(M\) which splits as \(M = N \cup_f -N\), where \(N\) is an oriented manifold with boundary and \(f: \partial N \to \partial N\) is an orientation preserving diffeomorphism, is called a twisted double of \(N\). The sphere splits naturally as the (un)twisted double of \(D^n\), as well as the twisted double of \(S^k \times D^{k+1}\) in the odd dimensional case. An interesting decomposition of \(S^4\) occurs as the twisted double of a regular neighbourhood of the standard embedded projective plane. This decomposition has been studied by a number of authors (see \textit{W. S. Massey} [Geom. Dedicata 2, 371-374 (1973; Zbl 0273.57019)], \textit{N. H. Kuiper} [Math. Ann. 208, 175-177 (1974; Zbl 0265.52002)], \textit{T. M. Price} [J. Aust. Math. Soc., Ser. A 23, 112-128 (1977; Zbl 0423.57006)], and the reviewer [Proc. Am. Math. Soc. 86, 328-330 (1982; Zbl 0505.57001)]) -- it is closely related to the structure of \(S^4\) as the quotient of \(CP^2\) under complex conjugation, or, equivalently, \(CP^2\) as the branched cover of \(S^4\) with branching set the embedded \(RP^2\). In the paper under review, the author shows that this example belongs in a family of related examples where \(S^4\) decomposes as a twisted double \(S^4 = N_m \cup -N_m\) of a regular neighbourhood of an embedded pseudoprojective plane \(P_m\) formed from the disk \(D^2\) by identifying boundary points which differ by \(e^{2\pi i k/m}\). A handle decomposition and corresponding Kirby calculus picture is given for \(N_m\) and its boundary \(Q_m\) is studied. \(Q_m\) is a rational homology sphere with fundamental group \(\langle\alpha,\beta|\alpha^n= \beta^n= (\alpha\beta)^n\rangle\). A generalization of the branched cover of \(S^4\) branched over \(RP^2\) is also described. This paper generalizes the splitting of \(S^4\) via a neighbourhood of \(RP^2\) by replacing \(RP^2\) by the pseudoprojective plane -- for the related question of when any sphere \(S^n\) can decompose as a …
Mathematics Subject Classification ID: 57N13
zbMATH DE Number: 1054587
zbMATH Keywords: pseudoprojective plane
zbMATH Keywords: twisted double
zbMATH Keywords: regular neighbourhood
reviewed by: Terry Lawson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270042395
OpenAlex ID: W2094304684
DOI: 10.3836/TJM/1270042395
description: scientific article; zbMATH DE number 1054587