Cohomogeneity one topological manifolds revisited

Abstract

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply connected cohomogeneity one topological manifolds in dimensions 5, 6, and 7 and obtain topological characterizations of these spaces. In these dimensions, these manifolds are homeomorphic to smooth manifolds.

Cohomogeneity one Alexandrov spaces in low dimensions

“…It is well-known that these manifolds admit invariant Riemannian metrics and are therefore Alexandrov spaces of cohomogeneity one. In the topological category, the corresponding classification results in dimensions at most 7 follow from combining the smooth classification with the classification of closed, simply connected cohomogeneity one topological manifolds with a non-smooth cohomogeneity one action in dimensions at most 7, obtained in [16]. It was also shown in [16] that closed, simply connected cohomogeneity one topological manifolds decompose as double cone bundles whose fibers are cones over spheres or the Poincaré homology sphere, and hence they admit invariant Alexandrov metrics.…”

“…We point out that the diagrams (G, H, K − , K + ) in Tables 1, 2, 3, 4, 5, 6, and 7 contain, as particular cases, the diagrams of non-smoothable cohomogeneity one actions on closed, simply connected topological manifolds in [16]; in this special situation the positively curved homogeneous spaces K ± ∕H are either spheres or the Poincaré homology sphere. Compared to the smooth and topological cases, the number of closed, simply connected cohomogeneity one Alexandrov spaces that are not manifolds increases substantially, due to the fact that at least one of the positively curved homogeneous spaces K ± ∕H is no longer a sphere or the Poincaré homology sphere.…”

“…In this subsection we collect basic facts on cohomogeneity one Alexandrov spaces and prove some preliminary results that we will use in the proof of Theorem A. For cohomogeneity one actions on smooth or topological manifolds, we refer the reader to [16,22], respectively.…”

“…In this context, a space with an effective action of a compact Lie group is of cohomogeneity one if its orbit space is one-dimensional. In the topological and smooth categories, the geometry and topology of cohomogeneityone manifolds have been studied extensively (see, for example, [16,36] and references therein) and closed, simply connected cohomogeneity-one manifolds of dimension at most 7 have been classified (see [16,22,[26][27][28]30]).…”

Equivariant cohomology of cohomogeneity-one actions: The topological case

“…We generalize in this way our previous result concerning smooth actions.Let M be a closed topological manifold, G a compact connected Lie group, and G × M → M a continuous action of cohomogeneity one, i.e., such that the orbit space M/G is one-dimensional. A complete description of such group actions has been obtained only recently by Galaz-García and Zarei in [5]. As a consequence, they were able to identify within the setup above the actions which do not fit into the smooth category.…”