Novikov's theorem in higher dimensions?

TL;DR

The authors address whether Novikov-type rigidity extends to higher dimensions by constructing a 5-dimensional manifold $X^5$ with a smooth foliation and a strong symplectic form that violate the natural analogue of Novikov's theorem. They develop foliated symplectic Lefschetz fibrations as the core tool, adapting Gompf's construction to foliated bases and proving a local model near singular fibers with controlled fiber area. The main result is a counterexample: a strong symplectic foliation for which the map $\pi_1(\,\mathcal{F})\to\pi_1(X)$ is not injective and a transversal loop can be contractible in $X$, underscoring that rigidity in higher dimensions differs from the 3-manifold case. The work suggests a weaker Novikov-type constraint may hold in odd dimensions and highlights the continued rigidity of strong symplectic foliations, while contrasting them with weak symplectic foliations. Overall, the paper clarifies the limits of extending 3D rigidity phenomena to higher-dimensional foliations and introduces new foliated Lefschetz techniques to construct and analyze such counterexamples.

Abstract

Novikov's theorem is a rigidity result on the class of taut foliations on three-manifolds. For higher dimensional manifolds, foliations with a strong symplectic form have been suggested as the class of foliations having similar rigidity properties to taut foliations on three-manifolds. This leads to the natural question of whether strong symplectic foliations satisfy an analogue of Novikov's theorem. In this paper, we construct a five-dimensional manifold with a smooth foliation and a strong symplectic form that does not satisfy the expected analogue of Novikov's theorem. Our example is a foliated Lefschetz fibration.

Figures (5)

• Figure 1: Reeb foliation on the solid torus.
• Figure 2: Construction of symplectic form near a singular fiber.
• Figure 3: Almost horizontal foliation on a strip $[-1,1] \times \mathbb{R}$.
• Figure 4: Left: A non-compact leaf $L$ in the base $B$. The monodromy around the singular point is equal to the difference in the monodromies around $\beta_u$ and $\beta_l$. Right: As the loop $\beta_u$ resp. $\beta_l$ is moved to the end of the cylinder, it limits to a loop $\beta_u'$ resp. $\beta_l'$ in the compact leaf $x=1$ resp. $x=0$. The monodromy map is trivial on $\beta_u$, $\beta_u'$. On the loops $\beta_l$, $\beta_l'$, the monodromy map is a negative Dehn twist about $V$.
• Figure 5: Cut and shear shown in a plane transverse to $W$. In $(X,\mathcal{F}')$, $\gamma'$ is a transversal.

Theorems & Definitions (18)

• Proposition 1.1
• Theorem 1
• Conjecture 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof : Proof of Lemma \ref{['lem:singsymp']}
• Definition 2.4
• Definition 2.5
• Proposition 2.6