Collapsing, solvmanifolds and infrahomogeneous spaces

Abstract

When phrased in terms of Hausdorff convergence, M. Gromov's almost flat manifold theorem states that if a compact manifold M admits a bounded curvature collapse to a point, then a finite cover of M is necessarily diffeomorphic to a nilmanifold. It is then tempting to ask whether the prescription of more general Hausdorff limits will still place some kind of homogeneity conditions or other severe restrictions on a manifold, or, in fact, whether a compact manifold can even be topologically characterized by the bounded curvature collapses it allows. In this paper, we study these questions in mainly the solvable category. Improving earlier works on this problem, and solving a conjecture of Fukaya, we first show that if a compact manifold M admits a bounded curvature collapse to a compact flat orbifold of arbitrary dimension, then a finite cover of M is diffeomorphic to a solvmanifold. As a partial converse to this result, we obtain the theorem that any compact infrasolvmanifold M admits a generalized Seifert fibration over a compact flat orbifold, and a sequence of locally homogeneous metrics such that M allows a bounded curvature collapse to this orbifold. Locally homogeneous collapsing metrics with bounded curvature and diameter are also constructed on any infrahomogeneous space which is modelled on a Lie goup whose radical is nontrivial. In the topological category, the above results already lead to a metrical characterization of infrasolvmanifolds in the spirit of Farrell and Hsiang. However, in the smooth setting, the situation is quite different, and for an analogous smooth characterization our first theorem would have to be appropiately sharpened. Indeed, we show that the classes of compact smooth manifolds that admit finite coverings by homogeneous spaces are in general strictly bigger than the corresponding classes of infrahomogeneous spaces. In the standard literature on the almost flat manifold theorem, this crucial distinction has nearly never been made. Finally, we investigate some stability and closedness properties of classes of infrasolvmanifolds under Hausdorff convergence, give an account of the history of the results presented here, and discuss some related open questions and conjectures.