A generalized Conner--Floyd conjecture and the immersion problem for low 2-torsion lens spaces
This paper gives new non-immersion results for \(2^k\)-torsion lens spaces, for \(k\) small, into Euclidean space. Let \(\alpha(d)\) denote the number of ones in the binary expansion of \(d\). The main result of the paper is the following theorem: ``For \(1\leq k\leq\alpha(d)\) the lens space \(L(2^k)^{2(d+\alpha(d)- K)+ 1}\) cannot be immersed into a Euclidean space of dimension \(4d-2\alpha(d)\) if either one of the following conditions hold: (i) \(\alpha(d)- k\leq 6\), (ii) \(2^{v(d)}> \alpha(d)-k\), where \(v(d)\) is the highest power of 2 dividing \(d\).'' Results for \(k\geq \alpha(d)\) were obtained in [\textit{J. Gonzalez}, J. Lond. Math. Soc. 63, No. 1, 247-256 (2001; Zbl 1013.57017)]. Furthermore, the above result for \(k=1\) corresponds to Davis' strong non-immersion theorem for real projective spaces. The paper uses the Brown-Peterson spectrum. The non-immersion is obtained by showing that the Euler class of a certain bundle \(n\) is non-trivial. Such obstructions are BP\(_*\)-multiples of the toral class \(\gamma_n\). So, the annihilator ideal \(I_n= \text{Ann}(\gamma_n)\) plays an important role. The paper contains a good description of the known results and, compared with the techniques used in related works, the simple theoretical approach (based on BP-techniques) suggests one should expect a similar type of solution in other cases.
scientific article; zbMATH DE number 1900978