Parallelizability of 4-dimensional infrasolvmanifolds

Abstract

We show that if $M$ is an orientable 4-dimensional infrasolvmanifold and either $\beta ={\beta }_1(M;\mathbb{Q})\ge 2$ or $M$ is a $\mathbb{S}o{l}_0^4$- or a $\mathbb{S}o{l}_{m,n}^4$-manifold (with $m\ne n$) then $M$ is parallelizable. There are non-parallelizable examples with $\beta =1$ for each of the other solvable Lie geometries ${\mathbb{E}}^4$, $\mathbb{N}i{l}^4$, $\mathbb{N}i{l}^3\times {\mathbb{E}}^1$ and $\mathbb{S}o{l}^3\times {\mathbb{E}}^1$. We also determine which non-orientable flat 4-manifolds have a $Pi{n}^{+}$- or $Pi{n}^{-}$-structure, and consider briefly this question for the other cases.

arXiv commentThis paper has been withdrawn by the author, as Lee and Thoung have shown that one of the claimed consequences is false

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