Decomposition of \(S^ 4\) as a twisted double of a certain manifold (Q1365385)

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 twisted double of regular neighbourhood of an embedded codimenson \(r\) submanifold, see the papers by \textit{W. S. Massey} [Indiana Univ. Math. J. 23, 791-812 (1974; Zbl 0285.57016)], \textit{H. F. Münzner} [Math. Ann. 251, 57-71 (1980; Zbl 0417.53030); 256, 215-232 (1981; Zbl 0438.53050)], and the reviewer [Houston J. Math. 8, 205-220 (1982; Zbl 0507.57017)].