Geometry & Topology Volume 18, issue 3 (2014)

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

Nonnegatively curved $5$–manifolds with almost maximal symmetry rank

Fernando Galaz-Garcia and Catherine Searle

Geometry & Topology 18 (2014) 1397–1435

DOI: 10.2140/gt.2014.18.1397

Bibliography

1	D Barden, Simply connected 5–manifolds, Ann. of Math. 82 (1965) 365 MR0184241
2	G E Bredon, Introduction to compact transformation groups, 46, Academic Press (1972) MR0413144
3	D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
4	Y Burago, M Gromov, G Perelman, AD Aleksandrov’s spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3, 222 MR1185284
5	T Chang, T Skjelbred, Group actions on Poincaré duality spaces, Bull. Amer. Math. Soc. 78 (1972) 1024 MR0307226
6	J Cheeger, D G Ebin, Comparison theorems in Riemannian geometry, AMS Chelsea Publishing (2008) MR2394158
7	J DeVito, The classification of simply connected biquotients of dimension at most 7 and 3 new examples of almost positively curved manifolds, PhD thesis, University of Pennsylvania (2011)
8	R Diestel, Graph theory, 173, Springer (2005) MR2159259
9	J Dinkelbach, B Leeb, Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3–manifolds, Geom. Topol. 13 (2009) 1129 MR2491658
10	H Duan, C Liang, Circle bundles over 4–manifolds, Arch. Math. (Basel) 85 (2005) 278 MR2172386
11	R Fintushel, Circle actions on simply connected 4–manifolds, Trans. Amer. Math. Soc. 230 (1977) 147 MR0458456
12	R Fintushel, Classification of circle actions on 4–manifolds, Trans. Amer. Math. Soc. 242 (1978) 377 MR496815
13	F Galaz-Garcia, Nonnegatively curved fixed point homogeneous manifolds in low dimensions, Geom. Dedicata 157 (2012) 367 MR2893494
14	F Galaz-Garcia, M Kerin, Cohomogeneity–two torus actions on nonnegatively curved manifolds of low dimension, Math. Z. 276 (2014) 133 MR3150196
15	F Galaz-Garcia, C Searle, Low-dimensional manifolds with nonnegative curvature and maximal symmetry rank, Proc. Amer. Math. Soc. 139 (2011) 2559 MR2784821
16	F Galaz-Garcia, W Spindeler, Nonnegatively curved fixed point homogeneous 5–manifolds, Ann. Global Anal. Geom. 41 (2012) 253 MR2876698
17	F Galaz-Garcia, W Spindeler, Erratum to : Nonnegatively curved fixed point homogeneous 5–manifolds, Ann. Global Anal. Geom. 45 (2014) 151 MR3165480
18	J C Gómez-Larrañaga, F González-Acuña, W Heil, S²– and P²–category of manifolds, Topology Appl. 159 (2012) 1052 MR2876711
19	J L Gross, J Yellen, editors, Handbook of graph theory, CRC Press (2004) MR2035186
20	K Grove, Geometry of, and via, symmetries, from: "Conformal, Riemannian and Lagrangian geometry", Univ. Lecture Ser. 27, Amer. Math. Soc. (2002) 31 MR1922721
21	K Grove, Developments around positive sectional curvature, from: "Geometry, analysis, and algebraic geometry" (editors H D Cao, S T Yau), Surv. Differ. Geom. 13, Int. Press (2009) 117 MR2537084
22	K Grove, C Searle, Positively curved manifolds with maximal symmetry rank, J. Pure Appl. Algebra 91 (1994) 137 MR1255926
23	K Grove, C Searle, Differential topological restrictions curvature and symmetry, J. Differential Geom. 47 (1997) 530 MR1617636
24	K Grove, B Wilking, A knot characterization and 1–connected nonnegatively curved 4–manifolds with circle symmetry, arXiv:1304.4827
25	B A Kleiner, Riemannian four-manifolds with nonnegative curvature and continuous symmetry, PhD thesis, University of California, Berkeley (1990)
26	D Montgomery, C T Yang, Groups on Sⁿ with principal orbits of dimension n − 3, Illinois J. Math. 4 (1960) 507 MR0125902
27	P S Mostert, On a compact Lie group acting on a manifold, Ann. of Math. 65 (1957) 447 MR0085460
28	W D Neumann, 3–dimensional G–manifolds with 2–dimensional orbits, from: "Proc. Conf. Transformation Groups", Springer (1968) 220 MR0245043
29	P Orlik, Seifert manifolds, 291, Springer (1972) MR0426001
30	P Orlik, F Raymond, Actions of SO(2) on 3–manifolds, from: "Proc. Conf. Transformation Groups", Springer (1968) 297 MR0263112
31	P S Pao, Nonlinear circle actions on the 4–sphere and twisting spun knots, Topology 17 (1978) 291 MR508892
32	G P Paternain, J Petean, Minimal entropy and collapsing with curvature bounded from below, Invent. Math. 151 (2003) 415 MR1953264
33	A V Pavlov, Five-dimensional biquotients of Lie groups, Sibirsk. Mat. Zh. 45 (2004) 1323 MR2123295
34	G Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159
35	G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math.DG/0307245
36	G Perelman, Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109
37	G Perelman, Alexandrov’s spaces with curvatures bounded from below, II, preprint (1991)
38	F Raymond, Classification of the actions of the circle on 3–manifolds, Trans. Amer. Math. Soc. 131 (1968) 51 MR0219086
39	X Rong, Positively curved manifolds with almost maximal symmetry rank, from: "Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II" (2002) 157 MR1950889
40	P Scott, The geometries of 3–manifolds, Bull. London Math. Soc. 15 (1983) 401 MR705527
41	C Searle, D Yang, On the topology of nonnegatively curved simply connected 4–manifolds with continuous symmetry, Duke Math. J. 74 (1994) 547 MR1272983
42	S Smale, On the structure of 5–manifolds, Ann. of Math. 75 (1962) 38 MR0141133
43	W Spindeler, Fixpunkthomogene S¹–Wirkungen auf 5–Mannigfaltigkeiten nichtnegativer Krümmung (2009)
44	B Totaro, Cheeger manifolds and the classification of biquotients, J. Differential Geom. 61 (2002) 397 MR1979366
45	B Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math. 191 (2003) 259 MR2051400
46	B Wilking, Nonnegatively and positively curved manifolds, from: "Metric and comparison geometry" (editors J Cheeger, K Grove), Surv. Differ. Geom. 11, Int. Press (2007) 25 MR2408263
47	B Wilking, Group actions on nonnegatively and positively curved manifolds, Lecture notes, Münster (2010)
48	W Ziller, Examples of Riemannian manifolds with nonnegative sectional curvature, from: "Metric and comparison geometry" (editors J Cheeger, K Grove), Surv. Differ. Geom. 11, Int. Press (2007) 63 MR2408264