An infrasolvmanifold which does not bound

J.A.Hillman

Abstract

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 $\mathbb{N}i{l}^3$ or $\mathbb{S}o{l}^3\times {\mathbb{E}}^1$ bounds. However there are $\mathbb{S}o{l}_1^4$-manifolds which are not boundaries. The question remains open for $\mathbb{N}i{l}^3\times {\mathbb{E}}^1$-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.