## Parallelizability of 4-dimensional infrasolvmanifolds

### J.A.Hillman

#### Abstract

We show that if $$M$$ is an orientable 4-dimensional infrasolvmanifold and either $$\beta=\beta_1(M;Q)\geq2$$ or $$M$$ is a $$\mathbb{S}ol_0^4$$- or a $$\mathbb{S}ol_{m,n}^4$$-manifold (with $$m\not=n$$) then $$M$$ is parallelizable. There are non-parallelizable examples with $$\beta=1$$ for each of the other solvable Lie geometries $$\mathbb{E}^4$$, $$\mathbb{N}il^4$$, $$\mathbb{S}ol_1^4$$, $$\mathbb{N}il^3\times\mathbb{E}^1$$ and $$\mathbb{S}ol^3\times\mathbb{E}^1$$. We also determine which non-orientable flat 4-manifolds have a $$Pin^+$$- or $$Pin^-$$-structure, and consider briefly this question for the other cases.

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

: Primary 57M50; secondary 57R15.

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 Friday, May 13, 2011