PreprintParallelizability of 4dimensional infrasolvmanifoldsJ.A.HillmanAbstractWe show that if \(M\) is an orientable 4dimensional 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 nonparallelizable 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 nonorientable flat 4manifolds have a \(Pin^+\) or \(Pin^\)structure, and consider briefly this question for the other cases. Keywords: 4manifold, geometry, infrasolvmanifold, parallelizable, \(Pin\)structure, Spin.AMS Subject Classification: Primary 57M50; secondary 57R15.
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