Abstract. The foliation of a Morse form ω on a closed manifold M is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of M and ω. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of rk ω and Sing ω. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if ω has more centers than conic singularities then b1(M ) = 0 and thus the foliation has no minimal components and homologically non-trivial compact leaves, its folitation graph being a tree.
A subspace or subgroup is isotropic under a bilinear map if the restriction of the map on it is trivial. We study maximal isotropic subspaces or subgroups under skew-symmetric maps, and in particular the isotropy index-the maximum dimension of an isotropic subspace or maximum rank of an isotropic subgroup. For a smooth closed orientable manifold M , we describe the geometric meaning of the isotropic subgroups of the first cohomology group with different coefficients under the cup product. We calculate the corresponding isotropy index, as well as the set of ranks of all maximal isotropic subgroups, for the connected sum and the direct product of manifolds. Finally, we study the relationship of the isotropy index with the first Betti number and the co-rank of the fundamental group. We also discuss applications of these results to the topology of foliations.Mathematics Subject Classification: 15A63, 15A03, 58K65.
The numbers m(ω) of minimal components and c(ω) of homologically independent compact leaves of the foliation of a Morse form ω on a connected smooth closed oriented manifold M are studied in terms of the rst non-commutative Betti numberb 1 (M ).
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