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 number of ones in the dyadic expansion of $n$--, the ($2n+1$)-dimensional $2^e$-torsion lens space cannot be embedded in Euclidean space of dimension $4n-2α(n)+1$. A slightly restricted version of this fact holds for $e<α(n)$. We also give an inductive construction of Euclidean embeddings for $2^e$-torsion lens spaces. Some of our best embeddings are within one dimension of being optimal.