Classification, Cobordism, and Curvature of Four-Dimensional Infra-Solvmanifolds

Abstract

Crystallographic groups of solvable Lie groups generalize the crystallographic groups of Euclidean space. The quotient of a solvable Lie group G by the action of a torsion-free crystallographic group of G is an infra-solvmanifold of G. Infra-solvmanifolds are aspherical manifolds that generalize both closed flat manifolds and closed almost flat manifolds (that is, infra-nilmanifolds). Here we complete the classification of 4-dimensional infra-solvmanifolds by classifying torsion-free crystallographic groups of certain 4-dimensional solvable Lie groups. The classification also includes those crystallographic groups with torsion. We prove that every 4-dimensional infra-solvmanifold is the boundary of a compact 5-dimensional manifold by constructing an involution on certain 4-dimensional infra-solvmanifolds which is either free, or has 2-dimensional fixed set. The Ricci signatures (that is, signatures of the Ricci transformation) of 4-dimensional Lie groups have been classified. A Ricci signature can be realized on an infra-solvmanifold M of G if M is a compact isometric quotient of G, where G has left invariant metric with prescribed Ricci signature. We classify which Ricci signatures can be realized on certain 4-dimensional infra-solvmanifolds.