An infrasolvmanifold which does not bound

TL;DR

The paper investigates which 4-dimensional infrasolvmanifolds bound or do not bound, providing classifications and examples, and explores cobounding 5-manifolds and embeddings into Euclidean space.

Contribution

It classifies bounding properties of various 4-dimensional infrasolvmanifolds and constructs explicit cobounding 5-manifolds for most flat 4-manifolds.

Findings

Certain classes of infrasolvmanifolds always bound.

Non-orientable Sol_1^4-manifolds can fail to bound.

Most flat 4-manifolds admit simple cobounding 5-manifolds.

Abstract

Every 4-dimensional infrasolvmanifold $M$ with $\beta_1(M;\mathbb{Q})>0$ or which is flat or has one of the geometries $\mathbb{N}il^4$, $\mathbb{S}ol_{m,n}^4$, or $\mathbb{S}ol_0^4$ bounds. However there are non-orientable $\mathbb{S}ol_1^4$-manifolds which do not bound. The question remains open for $\mathbb{N}il^3\times\mathbb{E}^1$-manifolds. We also give simple cobounding 5-manifolds for all but five of the 74 flat 4-manifolds, and investigate which orientable flat 4-manifolds embed in $\mathbb{R}^5$.