CAT(0) 4-manifolds are Euclidean

We prove that a topological 4-manifold of globally non-positive curvature is homeomorphic to Euclidean space.

[1]  Koichi Nagano,et al.  Geodesically complete spaces with an upper curvature bound , 2018, Geometric and Functional Analysis.

[2]  A. Lytchak.  Rigidity of spherical buildings and joins , 2005 .

[3]  Fredric D. Ancel,et al.  Interiors of compact contractible $n$-manifolds are hyperbolic $(n\geq 5)$ , 1997 .

[4]  J. Eells,et al.  Homotopy negligible subsets , 1969 .

[5]  L. Husch.  Finding a boundary for a 3-manifold , 1969 .

[6]  Morton Brown,et al.  The monotone union of open $n$-cells is an open $n$-cell , 1961 .

[7]  Dušan D. Repovš.  The recognition problem for topological manifolds: a survey , 1994 .

[8]  R. Wilder.  Topology of Manifolds , 1949 .

[9]  Improvements of upper curvature bounds , 2019, 1910.05287.

[10]  J. Stallings.  The Piecewise-Linear Structure of Euclidean Space , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  W. J. R. Mitchell.  Defining the boundary of a homology manifold , 1990 .

[12]  H. Toruńczyk.  Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds , 1978 .

[13]  Arcwise Isometries,et al.  A Course in Metric Geometry , 2001 .

[14]  F. Raymond.  Local triviality for Hurewicz fiberings of manifolds , 1965 .

[15]  K. S. Sarkaria.  Hyperbolic Manifolds∗ , 2016 .

[16]  Koichi Nagano,et al.  Asymptotic topological regularity of CAT(0) spaces , 2021, Annals of Global Analysis and Geometry.

[17]  Paul Thurston.  CAT(0) 4-manifolds possessing a single tame point are Euclidean , 1996 .

[18]  Koichi Nagano,et al.  Topological regularity of spaces with an upper curvature bound , 2018, Journal of the European Mathematical Society.

[19]  M. Gromov,et al.  Hyperbolic Manifolds, Groups and Actions , 1981 .

[20]  Morton Brown.  Some applications of an approximation theorem for inverse limits , 1960 .

[21]  Parabolic isometries of CAT(0) spaces and CAT(0) dimensions (Perspectives of Hyperbolic Spaces II) , 2003, math/0308274.

[22]  Bakul Sathaye.  Link Obstruction to Riemannian smoothings of locally CAT(0) 4-manifolds , 2017, 1707.03433.

[23]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[24]  Jan Cristina.  Gromov-Hausdorff convergence of metric spaces , 2008 .

[25]  Paul Thurston.  4-dimensional Busemann G-space are 4-manifolds , 1996 .

[26]  Jyh-Yang Wu.  Topological regularity theorems for Alexandrov spaces , 1997 .

[27]  Karol Borsuk,et al.  Theory Of Retracts , 1967 .

[28]  Bruno Benedetti,et al.  Collapsibility of CAT(0) spaces , 2011, Geometriae Dedicata.

[29]  Dušan D. Repovš,et al.  The Bing-Borsuk and the Busemann conjectures ∗ , 2008, 0811.0886.

[30]  James W. Cannon,et al.  The recognition problem: What is a topological manifold? , 1978 .

[31]  Gerald S. Ungar.  Conditions for a mapping to have the slicing structure property , 1969 .

[32]  M. Kapovich,et al.  Quasi-isometries preserve the geometric decomposition of Haken manifolds , 1997 .

[33]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[34]  J.-F. Lafont,et al.  4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings , 2010, 1002.4235.

[35]  Peter Petersen.  A finiteness theorem for metric spaces , 1990 .

[36]  S. Ferry.  Strongly regular mappings with compact ${\rm ANR}$ fibers are Hurewicz fiberings. , 1978 .

[37]  Michael H. Freedman,et al.  The topology of four-dimensional manifolds , 1982 .

[38]  Dušan D. Repovš,et al.  Locally $G$-homogeneous Busemann $G$-spaces , 2011, 1105.1439.

[39]  R. Daverman,et al.  A ghastly generalized $n$-manifold , 1981 .

[40]  T. Januszkiewicz,et al.  Hyperbolization of polyhedra , 1991 .

[41]  Strong homotopy equivalence of 3-manifolds , 1967 .

[42]  Karim A. Adiprasito,et al.  CAT(0) metrics on contractible manifolds , 2015, 1512.06403.

[43]  Stephan Stadler.  An obstruction to the smoothability of singular nonpositively curved metrics on 4-manifolds by patterns of incompressible tori , 2013, 1312.2198.

[44]  D. Rolfsen.  Strongly convex metrics in cells , 1968 .

[45]  S. Ivanov,et al.  Rigidity of Busemann convex Finsler metrics , 2017, Commentarii Mathematici Helvetici.

[46]  Herbert Busemann,et al.  The geometry of geodesics , 1955 .

[47]  R. Ho.  Algebraic Topology , 2022 .

[48]  Affine functions on CAT (κ)-spaces , 2006 .

[49]  Geodesics in piecewise linear manifolds , 1976 .