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

Let alpha(d) denote the number of ones in the binary expansion of d. For 1 less than or equal to k less than or equal to alpha(d) we prove that the 2(d + alpha(d) - k) + 1-dimensional, 2(k)-torsion lens space does not immerse in a Euclidian space of dimension 4d - 2alpha(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 2(k) 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. (C) 2002 Elsevier Science Ltd. All rights reserved.