Volume 21, issue 2 (2021)

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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