An infrasolvmanifold which does not bound



Orientable 4-dimensional infrasolvmanifolds bound orientably. We show that every non-orientable 4-dimensional infrasolvmanifold \(M\) with \(\beta=\beta_1(M;\mathbb{Q})>0\) or with geometry \(\mathbb{N}il^3\) or \(\mathbb{S}ol^3\times\mathbb{E}^1\) bounds. However there are \(\mathbb{S}ol_1^4\)-manifolds which are not boundaries. The question remains open for \(\mathbb{N}il^3\times\mathbb{E}^1\)-manifolds. Any possible counter-examples have severely constrained fundamental groups. We also find simple cobounding 5-manifolds for all but five of the 74 flat 4-manifolds, and investigate which flat 4-manifolds embed in \(\mathbb{R}^n\), for \(n=5,6\) or \(7\).

Keywords: boundary, embedding, geometry, infrasolvmanifold, 4-manifold.

AMS Subject Classification: Primary 57R75.

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Thursday, June 23, 2011