Presence of minimal components in a Morse form foliation

Singular foliations for M-theory compactification

“…This case is generic in the sense that Morse one-forms form a dense open subset of the set of all closed one-forms belonging to the fixed cohomology class f -hence a form which satisfies equations (3.41) can be replaced by a Morse form by infinitesimally perturbing b. Singular foliations defined by Morse one-forms were studied in [21]- [29] and [52]- [57]. 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.…”

“…Furthermore [21,23], there exists a system of Z-linearly independent one-cycles γ i ∈ H 1 (M, Z) (i = 1, . .…”

“…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.…”

“…The Morse case is generic in the sense that such 1-forms constitute an open and dense subset of the set of all closed one-forms belonging to the cohomology class f. In the Morse case, the singular foliationF can be described using the foliation graph [16][17][18] associated to the corresponding decomposition of M (see [18][19][20] and [21]- [29]), which provides a combinatorial way to encode some important aspects of the foliation's topology -up to neglecting the information contained in the so-called minimal components of the decomposition, components which should possess an as yet unexplored non-commutative geometric description. This provides a far-reaching extension of the picture found in [8] for the everywhere non-chiral case U = M , a case which corresponds to the situation when the foliation graph is reduced to either a circle (when F has compact leaves, being a fibration over S 1 ) or to a single so-called exceptional vertex (when F has non-compact dense leaves, being a minimal foliation).…”

Close cohomologous Morse forms with compact leaves

“…By [4], there exist c(ω) homologically independent compact leaves γ i that generate H ω , i.e. (i) the set of all closed 1-forms that have a compact leaf (this set is non-empty); (ii) the set of all closed 1-forms that have at least c homologically independent compact leaves.…”