Geometry & Topology Volume 28, issue 7 (2024)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

$\mathrm{CAT}(0)$ $4$–manifolds are Euclidean

Alexander Lytchak, Koichi Nagano and Stephan Stadler

Geometry & Topology 28 (2024) 3285–3308

DOI: 10.2140/gt.2024.28.3285

Bibliography

1	K Adiprasito, B Benedetti, Collapsibility of CAT(0) spaces, Geom. Dedicata 206 (2020) 181 MR4091542
2	K A Adiprasito, L Funar, CAT(0) metrics on contractible manifolds, preprint (2015) arXiv:1512.06403
3	S B Alexander, R L Bishop, The Hadamard–Cartan theorem in locally convex metric spaces, Enseign. Math. 36 (1990) 309 MR1096422
4	S Alexander, V Kapovitch, A Petrunin, Alexandrov geometry: foundations, 236, Amer. Math. Soc. (2024)
5	F D Ancel, C R Guilbault, Interiors of compact contractible n–manifolds are hyperbolic (n ≥ 5), J. Differential Geom. 45 (1997) 1 MR1443329
6	F D Ancel, M W Davis, C R Guilbault, CAT(0) reflection manifolds, from: "Geometric topology, I" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 441 MR1470741
7	V N Berestovsky, Borsuk’s problem on metrization of a polyhedron, Dokl. Akad. Nauk SSSR 268 (1983) 273 MR0688238
8	V N Berestovsky, D M Halverson, D Repovš, Locally G–homogeneous Busemann G–spaces, Differential Geom. Appl. 29 (2011) 299 MR2795840
9	M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) MR1744486
10	M Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960) 478 MR0115157
11	M Brown, The monotone union of open n–cells is an open n–cell, Proc. Amer. Math. Soc. 12 (1961) 812 MR0126835
12	Y Burago, M Gromov, G Perelman, A D Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3 MR1185284
13	D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
14	H Busemann, The geometry of geodesics, Academic (1955) MR0075623
15	J W Cannon, The recognition problem: what is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978) 832 MR0494113
16	A Cavicchioli, F Hegenbarth, D Repovš, Higher-dimensional generalized manifolds: surgery and constructions, Eur. Math. Soc. (2016) MR3558558
17	R J Daverman, J J Walsh, A ghastly generalized n–manifold, Illinois J. Math. 25 (1981) 555 MR0630833
18	M W Davis, T Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991) 347 MR1131435
19	M Davis, T Januszkiewicz, J F Lafont, 4–Dimensional locally CAT(0)–manifolds with no Riemannian smoothings, Duke Math. J. 161 (2012) 1 MR2872552
20	J Eells Jr., N H Kuiper, Homotopy negligible subsets, Compos. Math. 21 (1969) 155 MR0253331
21	S Ferry, Strongly regular mappings with compact ANR fibers are Hurewicz fiberings, Pacific J. Math. 75 (1978) 373 MR0506197
22	S C Ferry, Alexander duality and Hurewicz fibrations, Trans. Amer. Math. Soc. 327 (1991) 201 MR1028308
23	M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357 MR0679066
24	K Fujiwara, T Shioya, S Yamagata, Parabolic isometries of CAT(0) spaces and CAT(0) dimensions, Algebr. Geom. Topol. 4 (2004) 861 MR2100684
25	M Gromov, Hyperbolic manifolds, groups and actions, from: "Riemann surfaces and related topics" (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 183 MR0624814
26	D M Halverson, D Repovš, The Bing–Borsuk and the Busemann conjectures, Math. Commun. 13 (2008) 163 MR2488667
27	A Hatcher, Algebraic topology, Cambridge Univ. Press (2002) MR1867354
28	S T Hu, Theory of retracts, Wayne State Univ. Press (1965) 234 MR0181977
29	L S Husch, T M Price, Finding a boundary for a 3–manifold, Ann. of Math. 91 (1970) 223 MR0264678
30	S Ivanov, A Lytchak, Rigidity of Busemann convex Finsler metrics, Comment. Math. Helv. 94 (2019) 855 MR4046007
31	M Kapovich, B Leeb, Quasi-isometries preserve the geometric decomposition of Haken manifolds, Invent. Math. 128 (1997) 393 MR1440310
32	B Krakus, Any 3–dimensional G–space is a manifold, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968) 737 MR0236856
33	L Kramer, On the local structure and the homology of CAT(κ) spaces and Euclidean buildings, Adv. Geom. 11 (2011) 347 MR2795430
34	A Lytchak, Rigidity of spherical buildings and joins, Geom. Funct. Anal. 15 (2005) 720 MR2221148
35	A Lytchak, K Nagano, Geodesically complete spaces with an upper curvature bound, Geom. Funct. Anal. 29 (2019) 295 MR3925112
36	A Lytchak, K Nagano, Topological regularity of spaces with an upper curvature bound, J. Eur. Math. Soc. 24 (2022) 137 MR4375449
37	A Lytchak, V Schroeder, Affine functions on CAT(κ)–spaces, Math. Z. 255 (2007) 231 MR2262730
38	A Lytchak, S Stadler, Improvements of upper curvature bounds, Trans. Amer. Math. Soc. 373 (2020) 7153 MR4155203
39	D R McMillan Jr., Strong homotopy equivalence of 3–manifolds, Bull. Amer. Math. Soc. 73 (1967) 718 MR0229243
40	W J R Mitchell, Defining the boundary of a homology manifold, Proc. Amer. Math. Soc. 110 (1990) 509 MR1019276
41	W J R Mitchell, D Repovš, The topology of cell-like mappings, Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988) 265 MR1122860
42	K Nagano, Asymptotic topological regularity of CAT(0) spaces, Ann. Global Anal. Geom. 61 (2022) 427 MR4376533
43	P Petersen V, A finiteness theorem for metric spaces, J. Differential Geom. 31 (1990) 387 MR1037407
44	P Petersen V, Gromov–Hausdorff convergence of metric spaces, from: "Differential geometry : Riemannian geometry" (editors R Greene, S T Yau), Proc. Sympos. Pure Math. 54, Amer. Math. Soc. (1993) 489 MR1216641
45	F Raymond, Local triviality for Hurewicz fiberings of manifolds, Topology 3 (1965) 43 MR0159337
46	D Repovš, The recognition problem for topological manifolds: a survey, Kodai Math. J. 17 (1994) 538 MR1296925
47	D Rolfsen, Strongly convex metrics in cells, Bull. Amer. Math. Soc. 74 (1968) 171 MR0226591
48	B Sathaye, Link obstruction to Riemannian smoothings of locally CAT(0) 4–manifolds, preprint (2017) arXiv:1707.03433
49	S Stadler, An obstruction to the smoothability of singular nonpositively curved metrics on 4–manifolds by patterns of incompressible tori, Geom. Funct. Anal. 25 (2015) 1575 MR3426062
50	J Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962) 481 MR0149457
51	D A Stone, Geodesics in piecewise linear manifolds, Trans. Amer. Math. Soc. 215 (1976) 1 MR0402648
52	P Thurston, 4–Dimensional Busemann G–spaces are 4–manifolds, Differential Geom. Appl. 6 (1996) 245 MR1408310
53	P Thurston, CAT(0) 4–manifolds possessing a single tame point are Euclidean, J. Geom. Anal. 6 (1996) 475 MR1471903
54	H Toruńczyk, Concerning locally homotopy negligible sets and characterization of l₂–manifolds, Fund. Math. 101 (1978) 93 MR0518344
55	G S Ungar, Conditions for a mapping to have the slicing structure property, Pacific J. Math. 30 (1969) 549 MR0250311
56	R L Wilder, Topology of manifolds, 32, Amer. Math. Soc. (1949) MR0029491
57	J Y Wu, Topological regularity theorems for Alexandrov spaces, J. Math. Soc. Japan 49 (1997) 741 MR1466363