Parallelizability of 4-dimensional infrasolvmanifolds



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.

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

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Friday, May 13, 2011