On the immersion problem for \(2^r\)-torsion lens spaces (Q1764677)

The authors obtain some nonimmersion results in Euclidean spaces for \(2^r\)-torsion lens spaces. For positive integers \(n\) and \(r\), let \(L^{2n+1}(2^r)\) be the (2n + 1) -dimensional \(2^r\)-torsion lens space, that is the orbit space of \(S^{2n+1}\subset\mathbb{C}^{n+1}\) by the restriction to \(\mathbb{Z}/2^r\) of the diagonal action of \(S^1\). Let \(I(n,r)\) denote the dimension of the smallest Euclidean space in which \(L^{2n+1}(2^r)\) can be immersed. In a previous paper [J. Lond. Math. Soc., II. Ser. 63, No. 1, 247--256 (2001; Zbl 1013.57017)] the first named author observes that the inequalities \[ I(n,1)\leq I(n, 2)\leq\cdots \leq I(n, r)\leq I(n, r + 1)\leq\cdots\leq I(n)+ 1, \] where \(I(n)= I(n,\infty)\) for convenience, can be used to derive information about \(I(n,r)\) from known immersions of complex projective spaces as well as known nonimmersions of real projective spaces. For example, the immersion \(\mathbb{C} P^1\subseteq\mathbb{R}^3\) and the nonimmersion \(\mathbb{R} P^3\nsubseteq \mathbb{R}^3\) imply \(I(1,4)= 4\) for \(I\leq r<\infty\) (whereas \(I(1)= 3\)). The purpose of the present paper is to use obstruction theory to make this comparison approach more effective and, at the same time, to shed additional light on the role of the 2-torsion in the immersion problem for lens spaces. Let \(\alpha(n)\) denote the number of 1's in the binary expansion of \(n\). There exist series of papers with results concerning the relation between \(I(n,r)\) and \(\alpha(n)\): [\textit{J. Adem} and \textit{S. Gitler}, Bol. Soc. Mat. Mex., II. Ser. 9, 37--50 (19964; Zbl 0166.16501)], [\textit{L. Astey}, \textit{D. M. Davis} and \textit{J. González}, Bol. Soc. Mat. Mex., III. Ser. 9, No. 1, 151--163 (2003; Zbl 1053.57019)], [\textit{M. C. Crabb}, Proc. R. Soc. Edinb., Sect. A 117, No. 1/2, 155--170 (1991; Zbl 0719.57017)], [\textit{D. Davis} and \textit{M. Mahowald}, Proc. Lond. Math. Soc., III. Ser. 35, 333--344 (1977; Zbl 0357.55024)], [\textit{S. Gitler}, Topology 7, 47--53 (1968; Zbl 0166.19406)], [\textit{A. D. Randall}, Trans. Am. Math. Soc. 147, 135--151 (1970; Zbl 0194.55501)], [\textit{B. J. Sanderson}, Proc. Lond. Math. Soc., III. Ser. 14, 137--153 (1964; Zbl 0122.41703)], [\textit{T. A. Shimkus}, Bol. Soc. Mat. Mex., III. Ser. 9, No. 2, 339--357 (2003; Zbl 1042.57015)]. In this paper the authors analyze the remaining cases with \(\alpha(n)= 3\). The obtained results are following: Theorem 1.1. For \(n\equiv 1\pmod 4\) with \(\alpha(n)= 3\), \(L^{2n+1}(4)\) does not immerse in \(\mathbb{R}^{4n-5}\). Theorem 1.2. For even \(n\) with \(\alpha(n)= 3\), \(L^{2n+1}(8)\) does not immerse in \(\mathbb{R}^{4n-4}\). Theorem 1.3. For \(n\equiv 0\pmod 4\) with \(\alpha(n)= 3\), \(L^{2n+1}(4)\) does not immerse in \(\mathbb{R}^{4n-6}\). The authors discuss which of these results are optimal. For example the nonimmersion in Theorem 1.1 is optimal. In addition the following consequences in the case \(\alpha(n)= 4\) are deduced: Corollary 1.4. For odd \(m\) with \(\alpha(m)= 4\), \(L^{2m+1}(8)\) does not immerse in \(\mathbb{R}^{4n-8}\). Corollary 1.5. For \(m\equiv 2\pmod 4\) with \(\alpha(m)=4\), \(L^{2m+1}(8)\) does not immerse in \(\mathbb{R}^{4n-12}\). The proofs are based on the analysis of modified Postnikov towers for lifting the relevant stable normal bundles and on the paper by \textit{D. Davis} and \textit{M. Mahowald} [loc. cit.].