عنوان

Leafwise Morse-Novikov Cohomological Invariants of Foliations

TL51902

انگلیسی

Leafwise Morse-Novikov Cohomological Invariants of Foliations

[Thesis]

Islam, Md Shariful

Richardson, Ken

Texas Christian University

2019

95 p.

Ph.D.

Texas Christian University

2019

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 in the context of singular distributions given by the kernel of differential forms, and foliations of manifold. The kernel of a d_ω closed form is involutive and hence gives a foliation of a manifold. A transversely oriented foliation of a Riemannian manifold uniquely determines leafwise Morse-Novikov cohomology groups, which are independent of the choice of metric in the sense that different metrics correspond to isomorphic groups. The relevant 1-form ω, which is always leafwise closed, can be chosen to be the mean curvature 1-form of the transverse distribution of the foliation. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincar´e duality. We also show that for general singular foliations, the isomorphism classes of the induced leafwise Morse-Novikov cohomology groups are foliated homotopy invariants.

Mathematics

Islam, Md Shariful

Richardson, Ken

Texas Christian University

p

[Thesis]

276903

a

Y