## 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 $$\mathbb{N}il^3$$ 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.

: Primary 57R75.

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 Thursday, June 23, 2011