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# Homology, Homotopy and Applications

## Volume 11 (2009)

### Number 2

### On the embedding dimension of 2-torsion lens spaces

Pages: 133 – 160

DOI: http://dx.doi.org/10.4310/HHA.2009.v11.n2.a7

#### Authors

#### 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 $2^e$-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.

#### Keywords

Euclidean embeddings of lens spaces; connective complex K-theory; Brown-Peterson theory; Euler class; modified Postnikov towers

#### 2010 Mathematics Subject Classification

19L41, 55S45, 57R40

Published 1 January 2009