Topological regularity of spaces with an upper curvature bound

Abstract

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.

Geodesically complete spaces with an upper curvature bound

“…In the continuation [LN18] of this paper, we prove, that in some interesting cases, the measure-theoretical and homotopy-theoretical stratifications described above can be upgraded, to provide control on the topological type of the spaces in question. For instance, GCBA spaces which arise as limits of Riemannian manifolds can be very well understood, similarly to [Kap02].…”

“…In the continuation [LN18] of the present paper, this result will be used to prove that strainer maps are local Hurewicz fibrations. Here, we just apply the homotopic stability results of [Pet90] and deduce, that if a fiber F −1 (t) is compact in V then all nearby fibers are homotopy equivalent to it.…”

Noncritical maps on geodesically complete spaces with curvature bounded above

“…The following two theorems from geometric topology, used in [6] for strainer maps, provide sufficient conditions for a map to be a Hurewicz fibration. Both are due to Unger [10] and based on Michael's selection theorem.…”

“…A separable, locally compact, locally geodesically complete space with curvature bounded above is called a GCBA space. Lytchak and Nagano [5], [6] recently published the theory of GCBA spaces. Their main technical tool is a strainer on a GCBA space (see below).…”

Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

“…We will require an additional property to our classes of metric spaces: the geodesically completeness of the bicombing, which is equivalent to the standard geodesically completeness assumption in case of CAT(0) or Busemann convex spaces. This is an assumption which is usually required, even in case of CAT(0)-spaces, in order to control much better the local and asymptotic geometry (see the foundational works of B. Kleiner [Kle99] and of A. Lytchak and K. Nagano [Nag18], [LN19], [LN20]). For instance in [CS20] we proved that, for complete and geodesically complete CAT(0)-spaces, a packing condition at some scale r yields explicit control of the packing condition at any scale.…”