Volume 21, issue 2 (2021)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 5, 2389–2970
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
Spherical complexities with applications to closed geodesics

Stephan Mescher

Algebraic & Geometric Topology 21 (2021) 1021–1074
Bibliography
1 A Abbondandolo, M Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006) 254 MR2190223
2 A Abbondandolo, M Schwarz, A smooth pseudo-gradient for the Lagrangian action functional, Adv. Nonlinear Stud. 9 (2009) 597 MR2560122
3 L Auslander, On curvature in Finsler geometry, Trans. Amer. Math. Soc. 79 (1955) 378 MR71833
4 W Ballmann, G Thorbergsson, W Ziller, Closed geodesics on positively curved manifolds, Ann. of Math. 116 (1982) 213 MR672836
5 W Ballmann, G Thorbergsson, W Ziller, Existence of closed geodesics on positively curved manifolds, J. Differential Geom. 18 (1983) 221 MR710053
6 V Bangert, Y Long, The existence of two closed geodesics on every Finsler 2–sphere, Math. Ann. 346 (2010) 335 MR2563691
7 D Bao, S S Chern, Z Shen, An introduction to Riemann–Finsler geometry, 200, Springer (2000) MR1747675
8 T Bartsch, Topological methods for variational problems with symmetries, 1560, Springer (1993) MR1295238
9 V Benci, Periodic solutions of Lagrangian systems on a compact manifold, J. Differential Equations 63 (1986) 135 MR848265
10 R Bott, G Segal, The cohomology of the vector fields on a manifold, Topology 16 (1977) 285 MR645730
11 G E Bredon, Topology and geometry, 139, Springer (1993) MR1224675
12 M Clapp, D Puppe, Critical point theory with symmetries, J. Reine Angew. Math. 418 (1991) 1 MR1111200
13 M Clapp, D Puppe, Critical point theory of symmetric functions and closed geodesics, Differential Geom. Appl. 6 (1996) 367 MR1422342
14 F R Cohen, On the Lusternik–Schnirelmann category of an iterated loop space, from: "Stable and unstable homotopy" (editors W G Dwyer, S Halperin, R Kane, S O Kochman, M E Mahowald, P S Selick), Fields Inst. Commun. 19, Amer. Math. Soc. (1998) 39 MR1622336
15 R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773 MR1942249
16 O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, 103, Amer. Math. Soc. (2003) MR1990857
17 H Duan, Y Long, W Wang, Two closed geodesics on compact simply connected bumpy Finsler manifolds, J. Differential Geom. 104 (2016) 275 MR3557305
18 E Fadell, S Husseini, A note on the category of the free loop space, Proc. Amer. Math. Soc. 107 (1989) 527 MR984789
19 E Fadell, S Husseini, Category weight and Steenrod operations, Bol. Soc. Mat. Mexicana 37 (1992) 151 MR1317569
20 M Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003) 211 MR1957228
21 M Farber, Topology of robot motion planning, from: "Morse theoretic methods in nonlinear analysis and in symplectic topology" (editors P Biran, O Cornea, F Lalonde), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer (2006) 185 MR2276952
22 M Farber, Invitation to topological robotics, Eur. Math. Soc. (2008) MR2455573
23 M Farber, M Grant, Symmetric motion planning, from: "Topology and robotics" (editors M Farber, R Ghrist, M Burger, D Koditschek), Contemp. Math. 438, Amer. Math. Soc. (2007) 85 MR2359031
24 M Farber, M Grant, Robot motion planning, weights of cohomology classes, and cohomology operations, Proc. Amer. Math. Soc. 136 (2008) 3339 MR2407101
25 L Fernández Suárez, P Ghienne, T Kahl, L Vandembroucq, Joins of DGA modules and sectional category, Algebr. Geom. Topol. 6 (2006) 119 MR2199456
26 R H Fox, On the Lusternik–Schnirelmann category, Ann. of Math. 42 (1941) 333 MR4108
27 U Frauenfelder, W J Merry, G P Paternain, Floer homology for magnetic fields with at most linear growth on the universal cover, J. Funct. Anal. 262 (2012) 3062 MR2885947
28 J C Gómez-Larrañaga, F González-Acuña, Lusternik–Schnirelmann category of 3–manifolds, Topology 31 (1992) 791 MR1191380
29 M Grant, S Mescher, Topological complexity of symplectic manifolds, Math. Z. 295 (2020) 667 MR4100027
30 P de la Harpe, Brouwer degree, domination of manifolds, and groups presentable by products, Bull. Manifold Atlas 2017 (2017)
31 S T Hu, Theory of retracts, Wayne State Univ. Press (1965) 234 MR0181977
32 J D S Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987) 403 MR870737
33 W Klingenberg, Lectures on closed geodesics, 230, Springer (1978) MR0478069
34 W Klingenberg, Riemannian geometry, 1, de Gruyter (1982) MR666697
35 D Kotschick, C Löh, Fundamental classes not representable by products, J. Lond. Math. Soc. 79 (2009) 545 MR2506686
36 D Kotschick, C Neofytidis, On three-manifolds dominated by circle bundles, Math. Z. 274 (2013) 21 MR3054316
37 A Kriegl, P W Michor, The convenient setting of global analysis, 53, Amer. Math. Soc. (1997) MR1471480
38 J L Loday, Free loop space and homology, from: "Free loop spaces in geometry and topology" (editors J Latschev, A Oancea), IRMA Lect. Math. Theor. Phys. 24, Eur. Math. Soc. (2015) 137 MR3444364
39 Y Long, H Duan, Multiple closed geodesics on 3–spheres, Adv. Math. 221 (2009) 1757 MR2522828
40 L Meier, A Hilbert manifold model for mapping spaces, preprint (2010)
41 F Mercuri, The critical points theory for the closed geodesics problem, Math. Z. 156 (1977) 231 MR474115
42 A Oancea, Morse theory, closed geodesics, and the homology of free loop spaces, from: "Free loop spaces in geometry and topology" (editors J Latschev, A Oancea), IRMA Lect. Math. Theor. Phys. 24, Eur. Math. Soc. (2015) 67 MR3444362
43 R S Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966) 1 MR189028
44 R S Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology 5 (1966) 115 MR259955
45 P Petersen, Riemannian geometry, 171, Springer (2016) MR3469435
46 H B Rademacher, Morse-Theorie und geschlossene Geodätische, Habilitationsschrift, Universität Bonn (1991)
47 H B Rademacher, Nonreversible Finsler metrics of positive flag curvature, from: "A sampler of Riemann–Finsler geometry" (editors D Bao, R L Bryant, S S Chern, Z Shen), Math. Sci. Res. Inst. Publ. 50, Cambridge Univ. Press (2004) 261 MR2132661
48 H B Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Ann. 328 (2004) 373 MR2036326
49 H B Rademacher, The second closed geodesic on Finsler spheres of dimension n > 2, Trans. Amer. Math. Soc. 362 (2010) 1413 MR2563734
50 Y B Rudyak, On category weight and its applications, Topology 38 (1999) 37 MR1644063
51 Y B Rudyak, F Schlenk, Lusternik–Schnirelmann theory for fixed points of maps, Topol. Methods Nonlinear Anal. 21 (2003) 171 MR1980143
52 A S Schwarz, The genus of a fibre space, Tr. Mosk. Mat. Obs. 11 (1962) 99 MR0151982
53 E H Spanier, Algebraic topology, McGraw-Hill (1966) MR0210112
54 J Strom, Modern classical homotopy theory, 127, Amer. Math. Soc. (2011) MR2839990
55 R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17 MR61823
56 W Wang, Multiple closed geodesics on positively curved Finsler manifolds, Adv. Nonlinear Stud. 19 (2019) 495 MR3985003
57 E Zehnder, Lectures on dynamical systems: Hamiltonian vector fields and symplectic capacities, Eur. Math. Soc. (2010) MR2664490