Geodesic
ON THE RELATIVE LUSTERNIK–SCHNIRELMANN CATEGORY WITH RESPECT TO A REAL COHOMOLOGY CLASS
Glasgow mathematical journal, Tome 53 (2011) no. 1, pp. 169-183

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we study a homotopy invariant cat(X, B, [ω]) on a pair (X, B) of finite CW complexes with respect to the cohomology class of a continuous closed 1-form ω. This is a generalisation of a Lusternik–Schnirelmann-category-type cat(X, [ω]), developed by Farber in [3, 4], studying the topology of a closed 1-form. This paper establishes the connection with the original notion cat(X, [ω]) and obtains analogous results on critical points and homoclinic cycles. We also provide a similar ‘cuplength’ lower bound for cat(X, B, [ω]).
DOI : 10.1017/S0017089510000595
Mots-clés : Primary: 55M30, Secondary: 58E05, 37C29 
LI, TIEQIANG; SCHÜTZ, DIRK. ON THE RELATIVE LUSTERNIK–SCHNIRELMANN CATEGORY WITH RESPECT TO A REAL COHOMOLOGY CLASS. Glasgow mathematical journal, Tome 53 (2011) no. 1, pp. 169-183. doi: 10.1017/S0017089510000595
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[1] 1.Braverman, M. and Silantyev, V., Kirwan-Novikov inequalities on a manifold with boundary, Trans. Amer. Math. Soc. 358 (2006), 3329–3361. Google Scholar | DOI

[2] 2.Cornea, O., Some properties of the relative Lusternik–Schnirelmann category, in Stable and unstable homotopy (Dwyer, W. G., Halperin, S., Kane, R., Kochman, S. O., Mahowald, M. E. and Selick, P. S., Editors) (Toronto, Canada, 1996), pp. 67–72. Google Scholar

[3] 3.Farber, M., Zeros of closed 1-forms, homoclinic orbits, and Lusternik–Schnirelmann theory, Topol. Methods Nonlinear Anal. 19 (2002), 123–152. Google Scholar

[4] 4.Farber, M., Lusternik–Schnirelmann theory and dynamics, in Lusternik–Schnirelmann category and related topics (Cornea, O., Lupton, G., Oprea, J. and Tanre, D., Editors) (South Hadley, MA, 2001), pp. 95–111, Contemp. Math. 316 (American Mathematical Society, Providence, RI, 2002). Google Scholar

[5] 5.Farber, M., Topology of closed one-forms, Mathematical surveys and monographs, vol. 108 (American Mathematical Society, Providence, RI, 2004). Google Scholar

[6] 6.Farber, M. and Schütz, D., Cohomological estimates for cat(X, ξ), Geom. Topol. 11 (2007), 1255–1288. Google Scholar

[7] 7.Farber, M. and Schütz, D., Moving homology classes to infinity. Forum Math. 19 (2) (2007), 281–296. Google Scholar | DOI

[8] 8.Farber, M. and Schütz, D., Homological category weights and estimates for cat1(X ξ), J. Eur. Math. Soc (JEMS) 10 (1) (2008), 243–266. Google Scholar

[9] 9.Hatcher, A., Algebraic toplogy (Cambridge University Press, Cambridge, UK, 2002). Google Scholar

[10] 10.Latschev, J., Flows with Lyapunov one-forms and a generalization of Farber's theorem on homoclinic cycles, Int. Math. Res. Not. (5) (2004), 239–247. Google Scholar | DOI

[11] 11.Li, T., Topology of closed 1-forms on manifolds with boundary. PhD thesis (Durham University, 2009). Google Scholar

[12] 12.Moyaux, P. M., Lower bounds for the relative Lusternik–Schnirelmann category, Manuscr. Math. 101 (4) (2000), 533–542. Google Scholar | DOI

[13] 13.Reeken, M., Stability of critical points under small perturbations, part I: Topological theory, Manuscr. Math. 7 (1972), 387–411. Google Scholar | DOI

[14] 14.Spanier, E., Algebraic topology (Springer-Verlag, New York, 1966). Google Scholar

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