Parallelizability of 4-dimensional infrasolvmanifolds

J.A.Hillman

Abstract

We show that if $M$ is an orientable 4-dimensional infrasolvmanifold and either $\beta ={\beta }_1(M;Q)\ge 2$ 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 ${\mathbb{E}}^4$, $\mathbb{N}i{l}^4$, $\mathbb{S}o{l}_1^4$, $\mathbb{N}i{l}^3\times {\mathbb{E}}^1$ 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.

Keywords: 4-manifold, geometry, infrasolvmanifold, parallelizable, $Pin$-structure, Spin.

AMS Subject Classification: Primary 57M50; secondary 57R15.