Cohomological estimates for cat(<i>X,ξ</i>)

Farber, Michael; Schütz, Dirk

doi:10.2140/gt.2007.11.1255

Cited by 6 publications

(26 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Compared to the Morse inequalities of a Morse closed 1-form, cat(X, ξ) is applicable to more degenerate conditions, but in general it is harder to compute. In [5,6] Farber and Schütz improve the previous results and give more detailed insights on this issue.…”

Section: Introductionsupporting

confidence: 57%

On the Relative Lusternik–schnirelmann Category With Respect to a Real Cohomology Class

Li¹,

Schütz²

2010

Glasgow Math. J.

Self Cite

View full text Add to dashboard Cite

Abstract. 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]).2010 Mathematics Subject Classification. Primary: 55M30; Secondary: 58E05, 37C29.1. Introduction. Michael Farber [3, 4] initiated a systematic study of a generalisation of the classical Lusternik-Schnirelmann category with respect to a real cohomology class ξ of degree 1, cat(X, ξ), on a finite CW complex X. In [3] the power of such a notion is demonstrated in the study of the topology of critical points and the existence of homoclinic cycles on a closed manifold. Compared to the Morse inequalities of a Morse closed 1-form, cat(X, ξ) is applicable to more degenerate conditions, but, in general, it is harder to compute. In [6,8] Farber and Schütz improve the previous results and give more detailed insights on this issue.In this paper, we generalise the controlled version of the above notion to the relative case on a finite CW pair (X, B), which coincides with the absolute one when the subset B is empty. In particular, Section 2 introduces the definition of this relative category cat(X, B, ξ), and in Section 3 we describe the immediate properties of the object. As a main result, we obtain the inequality relating the relative categories for the three pairs of a triple. We summarise this in the following theorem:

show abstract

Section: Introductionsupporting

confidence: 57%

On the Relative Lusternik–schnirelmann Category With Respect to a Real Cohomology Class

Li¹,

Schütz²

2010

Glasgow Math. J.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finally, we compute cat 1 (X, ξ ) for products of surfaces as function of the cohomology class ξ ∈ H 1 (X; R). We compare our results with the computations of the invariant cat(X, ξ ) completed in [9]. We conclude that cat 1 (X, ξ ) may exceed cat(X, ξ ) by an arbitrarily large amount.…”

Section: Introductionmentioning

confidence: 69%

“…In order to show that the difference between them can be arbitrarily large, we have to introduce a controlled version of cat 1 (X, ξ ) which behaves better under cartesian products. The following discussion is very similar to [9,Section 9]. Let ω be a continuous closed 1-form on a finite cell complex X.…”

Section: A Controlled Version Of Cat 1 (X ξ )mentioning

confidence: 99%

Homological category weights and estimates for $\mathrm{cat}^1(X,\xi)$

Farber¹,

Belolipetsky²

2008

J. Eur. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we study a new notion of category weight of homology classes developing further the ideas of E. Fadell and S. Husseini [3]. In the case of closed smooth manifolds the homological category weight is equivalent to the cohomological category weight of E. Fadell and S. Husseini but these two notions are distinct already for Poincaré complexes. An important advantage of the homological category weight is its homotopy invariance. We use the notion of homological category weight to study various generalizations of the Lusternik-Schnirelmann category which appeared in the theory of closed 1-forms and have applications in dynamics. Our primary goal is to compare two such invariants cat(X, ξ ) and cat 1 (X, ξ ) which are defined similarly with reversion of the order of quantifiers. We compute these invariants explicitly for products of surfaces and show that they may differ by an arbitrarily large quantity. The proof of one of our main results, Theorem 8, uses an algebraic characterization of homology classes z ∈ H i (X; Z) (whereX → X is a free abelian covering) which are movable to infinity ofX with respect to a prescribed cohomology class ξ ∈ H 1 (X; R). This result is established in Part II which can be read independently of the rest of the paper.

show abstract

“…In this section we discuss results of this type for cat(X, ξ) following our paper [24]; some lower bounds for cat(X, ξ) were obtained earlier in the papers [15] and [18]. We give in this section the main definitions, state principal results and illustrate them by several specific examples; however for complete proofs we refer the reader to our original papers [24], [25] and [23]. Let X be a finite polyhedron and ξ ∈ H 1 (X; R).…”

Section: Cohomological Estimates For Cat(x ξ)mentioning

confidence: 99%

“…where N ⊂X is a connected neighborhood of infinity with respect to χ, see §2 of [24] and Lemma 5.2 from [1].…”

mentioning

confidence: 99%

Closed 1-forms in topology and dynamics

Farber¹,

Schütz²

2008

Russ. Math. Surv.

Self Cite

View full text Add to dashboard Cite

Abstract. This article surveys recent progress of results in topology and dynamics based on techniques of closed one-forms. Our approach allows us to draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically we describe a Lusternik -Schnirelmann type theory for closed one-forms, the focusing effect for flows and the theory of Lyapunov one-forms. We also discuss recent results about cohomological treatment of the invariants cat(X, ξ) and cat 1 (X, ξ) and their explicit computation in certain examples. To S.P. Novikov on the occasion of his 70-th birthdayIn 1981 S.P. Novikov [36,39] initiated a generalization of Morse theory which gives topological estimates on the numbers of zeros of closed 1-forms, see also [41,42]. Novikov was motivated by a variety of important problems of mathematical physics, leading, in one way or another, to the problem of finding relations between the topology of the underlying manifold and the number of zeros which closed 1-forms in a specific one-dimensional real cohomology class possess. Recall that a closed 1-form can be viewed as a multi-valued function or functional whose branching behavior is fully characterized by the cohomology class. In [39] Novikov studies various problems of physics where the motion can be reduced to the principle of an extremal action S, which is a multi-valued functional on the space of curves, with the variation δS being a well-defined closed 1-form, cf. [38,37,40]. Among problems of this kind are the Kirchhoff equations for the motion of a rigid body in ideal fluid, the Leggett equation for the magnetic momentum, and others. His fundamental idea in [39] was based on a plan to construct a chain complex, now called the Novikov complex, which uses dynamics of the gradient flow in the abelian covering associated with the cohomology class. The dynamics of gradient flows appears traditionally in Morse theory providing a bridge between the critical set of a function and the global ambient topology.Date: May 31, 2018.

show abstract

Cohomological estimates for cat(X,ξ)

Cited by 6 publications

References 14 publications

On the Relative Lusternik–schnirelmann Category With Respect to a Real Cohomology Class

On the Relative Lusternik–schnirelmann Category With Respect to a Real Cohomology Class

Homological category weights and estimates for $\mathrm{cat}^1(X,\xi)$

Closed 1-forms in topology and dynamics

Contact Info

Product

Resources

About