Parallelizability of 4-dimensional infrasolvmanifolds

TL;DR

This paper investigates the conditions under which 4-dimensional infrasolvmanifolds are parallelizable, identifying specific geometric and topological criteria, and also explores the existence of Pin structures on non-orientable flat 4-manifolds.

Contribution

It characterizes parallelizability of 4D infrasolvmanifolds based on Betti number and geometry, and determines Pin structure existence on non-orientable flat 4-manifolds.

Findings

Certain infrasolvmanifolds are parallelizable based on Betti number and geometry.

Non-parallelizable examples exist with Betti number 1 in specific geometries.

Conditions for Pin^+ and Pin^- structures on non-orientable flat 4-manifolds are identified.

Abstract

We show that if $M$ is an orientable 4-dimensional infrasolvmanifold and either $\beta=\beta_1(M;\mathbb{Q})\geq2$ 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 $\mathbb{E}^4$, $\mathbb{N}il^4$, $\mathbb{N}il^3\times\mathbb{E}^1$ 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.