Rigidity of Lie affine foliations
Stephane Geudens
TL;DR
The paper analyzes rigidity of Lie foliations modeled on the non-abelian Lie algebra $\mathfrak{aff}(1)$ by linking deformation cohomology $H^{1}(\mathcal{A})$ to Morse–Novikov cohomology. It develops a precise description via a mapping-cone framework, and obtains explicit criteria for when $H^{1}(\mathcal{A})=0$, leading to rigidity results. A central tool is the computation of Morse–Novikov cohomology $H^{\bullet}_{\theta}(M)$ for nowhere-vanishing $\theta$ with discrete period group, reduced to Morse–Novikov data on mapping tori $\mathbb{T}_{\varphi}$ with $\theta=cdt$, yielding explicit kernels and cokernels of $[\varphi^{*}-e^{-c}\mathrm{Id}]$. The authors prove that any Lie affine foliation on a compact, connected, orientable 3- or 4-manifold is rigid when deformed as a Lie foliation, while higher-dimensional cases can fail to be rigid; model Lie affine foliations provide explicit, independent-cohomology criteria and illustrate the role of monodromy in rigidity.
Abstract
In previous work by El Kacimi Alaoui-Guasp-Nicolau, a cohomological criterion is given for a Lie $\mathfrak{g}$-foliation on a compact manifold to be rigid among nearby Lie foliations. Our aim is to look for examples of this rigidity statement in case the Lie foliation is modeled on the two-dimensional non-abelian Lie algebra $\mathfrak{g}=\mathfrak{aff}(1)$. We study the relevant cohomology group in detail, showing that it can be expressed in terms of Morse-Novikov cohomology. We find the precise conditions under which it vanishes, which yields many examples of rigid Lie affine foliations. In particular, we show that any Lie affine foliation on a compact, connected, orientable manifold of dimension $3$ or $4$ is rigid when deformed as a Lie foliation. Our results rely on a computation of the Morse-Novikov cohomology groups associated with a nowhere-vanishing closed one-form with discrete period group, which may be of independent interest.