Homotopy invariance of cohomology and signature of a Riemannian foliation

Habib, Georges; Richardson, Ken

doi:10.1007/s00209-018-2195-x

Cited by 3 publications

(2 citation statements)

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“…Much work on these cohomology groups has been done (c.f. [1], [11], [18], [2], [5] [6], and the associated references).…”

Section: Basic and Antibasic Cohomology Of Foliationsmentioning

confidence: 99%

New cohomological invariants of foliations

Habib,

Richardson

2019

Preprint

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Given a smooth foliation on a closed manifold, basic forms are differential forms that can be expressed locally in terms of the transverse variables. The space of basic forms yields a differential complex, because the exterior derivative fixes this set. The basic cohomology is the cohomology of this complex, and this has been studied extensively. Given a Riemannian metric, the adjoint of the exterior derivative maps the orthogonal complement of the basic forms to itself, and we call the resulting cohomology the "antibasic cohomology". Although these groups are defined using the metric, the dimensions of the antibasic cohomology groups are invariant under diffeomorphism and metric changes. If the underlying foliation is Riemannian, the groups are foliated homotopy invariants that are independent of basic cohomology and ordinary cohomology of the manifold. For this class of foliations we use the codifferential on antibasic forms to obtain the corresponding Laplace operator, develop its analytic properties, and prove a Hodge theorem. We then find some topological and geometric properties that impose restrictions on the antibasic Betti numbers.

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“…Much work on these cohomology groups has been done (c.f. [1], [11], [18], [2], [5] [6], and the associated references).…”

Section: Basic and Antibasic Cohomology Of Foliationsmentioning

confidence: 99%

New cohomological invariants of foliations

Habib,

Richardson

2019

Preprint

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“…K. Richardson and G. Habib used basic Morse-Novikov cohomology to prove that the basic signature and the Álvarez class of a Riemannian foliation are homotopy invariants [10]. They have also used a modified differential as in Morse-Novikov cohomology to define a twisted basic cohomology for Riemannian foliations that satisfies Poincaré duality [9]. J.…”

Section: Introductionmentioning

confidence: 99%

Morse-Novikov cohomology on foliated manifolds

Islam¹

2022

Preprint

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The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential d ω = d + ω∧, where ω is a closed 1-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincaré duality. The resulting isomorphisms yield a Hodge diamond structure for leafwise Morse-Novikov cohomology.

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