A generalized Conner–Floyd conjecture and the immersion problem for low 2-torsion lens spaces

Let α(d) denote the number of ones in the binary expansion of d. For 1⩽k⩽α(d) we prove that the 2(d+α(d)−k)+1-dimensional, 2k-torsion lens space does not immerse in a Euclidian space of dimension 4d−2α(d) provided certain technical condition holds. The extra hypothesis is easily eliminated in the case k=1 recovering Davis’ strong non-immersion theorem for real projective spaces. For k>1 this is a deeper problem (solved only in part) that requires a close analysis of the interaction between the Brown–Peterson 2-series and its 2k analogue. The methods are based on a partial generalization of the Brown–Peterson version for the Conner–Floyd conjecture used in this context to detect obstructions for the existence of Euclidian immersions.