On the embedding dimension of 2-torsion lens spaces

Abstract

Using the ku- and BP-theoretic versions of Astey's cobordism obstruction for the existence of smooth Euclidean embeddings of stably almost complex manifolds, we prove that, for e greater than or equal to α(n), the (2n+1)-dimensional 2e-torsion lens space cannot be embedded in Euclidean space of dimension 4n-2α(n)+1. (Here α(n) denotes the number of ones in the dyadic expansion of a positive integer n.) A slightly restricted version of this fact holds for e < α(n). We also give an inductive construction of Euclidean embeddings for 2e-torsion lens spaces. Some of our best embeddings are within one dimension of being optimal.