Homology, Homotopy and Applications
Volume 11 (2009)
On the embedding dimension of 2-torsion lens spaces
Pages: 133 – 160
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 $2^e$-torsion lens space cannot be embedded in Euclidean space of dimension $4n − 2 α(n) + 1$. (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 $2^e$-torsion lens spaces. Some of our best embeddings are within one dimension of being optimal.
Euclidean embeddings of lens spaces; connective complex K-theory; Brown-Peterson theory; Euler class; modified Postnikov towers
2010 Mathematics Subject Classification
19L41, 55S45, 57R40