Cohomological estimates for cat(X,ξ)

Farber, Michael¹ ; Schütz, Dirk¹

¹ Department of Mathematics, University of Durham, Durham DH1 3LE, UK

Geometry & topology, Tome 11 (2007) no. 3, pp. 1255-1288.

Résumé

This paper studies the homotopy invariant $cat\left(X,\xi \right)$ introduced in [1: Michael Farber, ‘Zeros of closed 1-forms, homoclinic orbits and Lusternik–Schnirelman theory’, Topol. Methods Nonlinear Anal. 19 (2002) 123–152]. Given a finite cell-complex $X$, we study the function $\xi \mapsto cat\left(X,\xi \right)$ where $\xi$ varies in the cohomology space $H^1\left(X;R\right)$. Note that $cat\left(X,\xi \right)$ turns into the classical Lusternik–Schnirelmann category $cat\left(X\right)$ in the case $\xi =0$. Interest in $cat\left(X,\xi \right)$ is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1–forms, see [1] and [2: Michael Farber, ‘Topology of closed one-forms’, Mathematical Surveys and Monographs 108 (2004)].

In this paper we significantly improve earlier cohomological lower bounds for $cat\left(X,\xi \right)$ suggested in [1] and [2]. The advantages of the current results are twofold: firstly, we allow cohomology classes $\xi$ of arbitrary rank (while in [1] the case of rank one classes was studied), and secondly, the theorems of the present paper are based on a different principle and give slightly better estimates even in the case of rank one classes. We introduce in this paper a new controlled version of $cat\left(X,\xi \right)$ and find upper bounds for it. We apply these upper and lower bounds in a number of specific examples where we explicitly compute $cat\left(X,\xi \right)$ as a function of the cohomology class $\xi \in H^1\left(X;R\right)$.

Détail

DOI : 10.2140/gt.2007.11.1255

Keywords: Lusternik–Schnirelmann theory, closed 1-form, cup-length

Bibliographie

[1] R Bieri, R Geoghegan, Connectivity properties of group actions on non-positively curved spaces, Mem. Amer. Math. Soc. 161 (2003)

[2] R Bieri, W D Neumann, R Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987) 451

[3] N Bourbaki, Elements of mathematics. Commutative algebra, Hermann (1972)

[4] O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, Mathematical Surveys and Monographs 103, Amer. Math. Soc. (2003)

[5] M Farber, Lusternik–Schnirelman theory and dynamics, from: "Lusternik–Schnirelmann category and related topics (South Hadley, MA, 2001)", Contemp. Math., Amer. Math. Soc. (2002) 95

[6] M Farber, Zeros of closed 1-forms, homoclinic orbits and Lusternik–Schnirelman theory, Topol. Methods Nonlinear Anal. 19 (2002) 123

[7] M Farber, Topology of closed one-forms, Mathematical Surveys and Monographs 108, Amer. Math. Soc. (2004)

[8] M Farber, T Kappeler, Lyusternik–Shnirel'man theory and dynamics. II, Tr. Mat. Inst. Steklova 247 (2004) 252

[9] M Farber, D Schütz, Closed 1-forms with at most one zero, Topology 45 (2006) 465

[10] M Farber, D Schütz, Moving homology classes to infinity, Forum Mathematicum 19 (2007) 281

[11] M Farber, D Schütz, Homological category weights and estimates for $\mathrm{cat}^1(X,\xi)$

[12] I M James, On category, in the sense of Lusternik–Schnirelmann, Topology 17 (1978) 331

[13] F Latour, Existence de $1$-formes fermées non singulières dans une classe de cohomologie de de Rham, Inst. Hautes Études Sci. Publ. Math. (1994)

[14] J Latschev, Flows with Lyapunov one-forms and a generalization of Farber's theorem on homoclinic cycles, Int. Math. Res. Not. (2004) 239

[15] G Levitt, $1$-formes fermées singulières et groupe fondamental, Invent. Math. 88 (1987) 635

[16] S P Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR 260 (1981) 31

[17] D Schütz, On the Lusternik–Schnirelman theory of a real cohomology class, Manuscripta Math. 113 (2004) 85

[18] E H Spanier, Algebraic topology, McGraw-Hill (1966)