Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T01:00:37.609Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE RELATIVE LUSTERNIK–SCHNIRELMANN CATEGORY WITH RESPECT TO A REAL COHOMOLOGY CLASS

Published online by Cambridge University Press:  25 August 2010

TIEQIANG LI  and
DIRK SCHÜTZ
TIEQIANG LI
Affiliation:
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK e-mail: tieqiang.li@durham.ac.uk, dirk.schuetz@durham.ac.uk
DIRK SCHÜTZ
Affiliation:
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK e-mail: tieqiang.li@durham.ac.uk, dirk.schuetz@durham.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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]).

Keywords

Primary: 55M30Secondary: 58E0537C29
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 1 , January 2011 , pp. 169 - 183
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Braverman, M. and Silantyev, V., Kirwan-Novikov inequalities on a manifold with boundary, Trans. Amer. Math. Soc. 358 (2006), 3329–3361.CrossRefGoogle Scholar
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.Farber, M., Zeros of closed 1-forms, homoclinic orbits, and Lusternik–Schnirelmann theory, Topol. Methods Nonlinear Anal. 19 (2002), 123–152.Google Scholar
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.Farber, M., Topology of closed one-forms, Mathematical surveys and monographs, vol. 108 (American Mathematical Society, Providence, RI, 2004).Google Scholar
6.Farber, M. and Schütz, D., Cohomological estimates for cat(X, ξ), Geom. Topol. 11 (2007), 1255–1288.Google Scholar
7.Farber, M. and Schütz, D., Moving homology classes to infinity. Forum Math. 19 (2) (2007), 281–296.CrossRefGoogle Scholar
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.Hatcher, A., Algebraic toplogy (Cambridge University Press, Cambridge, UK, 2002).Google Scholar
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.CrossRefGoogle Scholar
11.Li, T., Topology of closed 1-forms on manifolds with boundary. PhD thesis (Durham University, 2009).Google Scholar
12.Moyaux, P. M., Lower bounds for the relative Lusternik–Schnirelmann category, Manuscr. Math. 101 (4) (2000), 533–542.CrossRefGoogle Scholar
13.Reeken, M., Stability of critical points under small perturbations, part I: Topological theory, Manuscr. Math. 7 (1972), 387–411.CrossRefGoogle Scholar
14.Spanier, E., Algebraic topology (Springer-Verlag, New York, 1966).Google Scholar