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

Jesús González (Department of Mathematics, CINVESTAV–IPN, Mexico City, Mexico)

Peter Landweber (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Thomas Shimkus (Department of Mathematics, University of Scranton, Pennsylvania, U.S.A.)

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

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