TAILIEUCHUNG - Sufficient conditions for the compactifiability of a closed one-form foliation
In this paper, we give sufficient conditions for compactifi of the foliation in homological terms. We also show that under these conditions, the foliation can be defined by closed 1 -forms with the ranks of their groups of periods in a certain range. | Turk J Math (2017) 41: 1344 – 1353 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Sufficient conditions for the compactifiability of a closed one-form foliation Irina GELBUKH∗ Computing Research Center (CIC), National Polytechnic Institute, Mexico City, Mexico Received: • Accepted/Published Online: • Final Version: Abstract: We study the foliation defined by a closed 1 -form on a connected smooth closed orientable manifold. We call such a foliation compactifiable if all its leaves are closed in the complement of the singular set. In this paper, we give sufficient conditions for compactifiability of the foliation in homological terms. We also show that under these conditions, the foliation can be defined by closed 1 -forms with the ranks of their groups of periods in a certain range. In addition, we describe the structure of the group generated by the homology classes of all compact leaves of the foliation. Key words: Closed one-form foliation, compact leaves, form’s rank 1. Introduction Consider a closed 1-form ω on a connected smooth closed orientable n-dimensional manifold M ; denote by Sing ω the set of its singularities. On M \ Sing ω , this form defines a codimension-one foliation Fω . Such foliations have important applications in modern physics, for example, in the theory of supergravity [2, 3]. Compact foliations, that is, foliations that consist entirely of leaves closed in M , . compact, are well studied. However, the property of compactness of a foliation is too restrictive: say, manifolds that admit a compact foliation defined by a Morse form (locally the differential of a Morse function) are sphere S n and bundle over S 1 (Proposition ). In addition, compactness is easily destroyed by a local perturbation of the form, for example, by adding a local center – the trivial center-saddle pairing [4]. Instead, we study a weaker but more
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