Topology of Closed One-Forms

Farber, Michael

doi:10.1090/surv/108

Cited by 124 publications

(192 citation statements)

References 6 publications

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“…The backgrounds discussed in this paper display a rich interplay between spin geometry, the theory of G-structures, the theory of foliations and the topology of closed one-forms [58]. This suggests numerous problems that could be approached using the methods and results of reference [8] and of this paper -not least of which concerns the generalization to the case of singular foliations of the non-commutative geometric description of the leaf space.…”

Section: Conclusion and Further Directionsmentioning

confidence: 96%

“…Let Π f = im(per f ) ⊂ R be the period group of the cohomology class f and ρ(f) = rkΠ f be its irrationality rank. The general results summarized in the following subsection hold for any smooth, compact and connected manifold of dimension d which is strictly bigger than two, under the assumption that the set of zeroes of ω (which in Novikov theory [58] is called the set of singular points):…”

Section: Jhep03(2015)116mentioning

confidence: 99%

“…One can describe [18,58] the singular foliationF ω of M defined by ω as the partition of M induced by the equivalence relation ∼ defined as follows. We put p ∼ q if there exists a smooth curve γ : [0, 1] → M such that:…”

Section: Jhep03(2015)116mentioning

confidence: 99%

“…What we shall call the "Novikov decomposition" is a generalization of the Morse decomposition [59][60][61], which was introduced in [19,20] (see also [18,57]) and used extensively in [21]- [29]; the name is motivated by analogy with "Morse decomposition", due to the role which this decomposition plays in the modern study of the topology of closed oneforms [58]. Define C max to be the union of all compact leaves and C min to be the union of all non-compactifiable leaves of F ω ; it is clear that these two subsets of M are disjoint.…”

Section: The Novikov Decomposition Of Mmentioning

confidence: 99%

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Singular foliations for M-theory compactification

Babalic

Lazaroiu

2015

J. High Energ. Phys.

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Abstract:We use the theory of singular foliations to study N = 1 compactifications of eleven-dimensional supergravity on eight-manifolds M down to AdS 3 spaces, allowing for the possibility that the internal part ξ of the supersymmetry generator is chiral on some locus W which does not coincide with M . We show that the complement M \ W must be a dense open subset of M and that M admits a singular foliationF endowed with a longitudinal G 2 structure and defined by a closed one-form ω, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet W. When ω is a Morse form, the chiral locus is a finite set of points, consisting of isolated zero-dimensional leaves and of conical singularities of seven-dimensional leaves. In that case, we describe the topology ofF using results from Novikov theory. We also show how this description fits in with previous formulas which were extracted by exploiting the Spin(7) ± structures which exist on the complement of W.

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Section: Conclusion and Further Directionsmentioning

confidence: 96%

Section: Jhep03(2015)116mentioning

confidence: 99%

Section: Jhep03(2015)116mentioning

confidence: 99%

Section: The Novikov Decomposition Of Mmentioning

confidence: 99%

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Singular foliations for M-theory compactification

Babalic

Lazaroiu

2015

J. High Energ. Phys.

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“…The list of such topics includes (a) constructions of chain complexes (more general than the Novikov complex) which are able to capture the link between topology of the manifold and topology of the set of zeros, (b) various types of inequalities for closed 1-forms (with Morse or Bott type nondegeneracy assumptions), (c) equivariant inequalities and (d) problems about sharpness of these inequalities. Two recent monographs [18], [47] give expositions of Novikov theory from quite different angles. Topology of closed 1-forms is a broader research area which together with the Novikov theory studies a new Lusternik -Schnirelmann type theory for closed 1-forms initiated in 2002 in [15].…”

Section: To Sp Novikov On the Occasion Of His 70-th Birthdaymentioning

confidence: 99%

Closed 1-forms in topology and dynamics

Farber¹,

Schütz²

2008

Russ. Math. Surv.

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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.

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