2012/08/31 by Igor Belegradek
Mathematics · #Advanced Operator Algebra Research #Codimension #Contractible space #Covering space #Diffeomorphism #Geometric Analysis and Curvature Flows #Homotopy #Homotopy and Cohomology in Algebraic Topology #Manifold (fluid mechanics) #Metric (unit) #Riemannian manifold #Sectional curvature #Simply connected space #math.DG #math.GR #math.GT #msc:53C21 #msc:57N15 #msc:57S30
paper · pdf · doi:10.1112/plms/pdu021
29 pages
openalex publication_date 2014/06/12 · arxiv created 2014/08/03 · openalex created_date 2016/06/24 · arxiv updated 2017/05/17 · openalex updated_date 2026/08/05
We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function, and for such G we prove the following: (1) every open complete nonpositively curved Riemannian K ( G , 1 ) manifold that is homotopy equivalent to a finite complex of codimension at least 3 is an open regular neighborhood of a subcomplex of the same codimension; (2) each tangential homotopy type contains infinitely many open K ( G , 1 ) manifolds that admit no complete nonpositively curved metric even though their universal cover is the Euclidean space. A sample application is that an open contractible manifold W is homeomorphic to a Euclidean space if and only if W × S 1 admits a complete Riemannian metric of nonpositive curvature.