Miscellaneous — Question 15.2

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A codimension-one foliation $\mathcal F$ decomposes a $3$-manifold $M$ into surface leaves, locally as the horizontal slices of a product chart. In the terminology of the source, it is taut if a closed curve transverse to $\mathcal F$ meets every leaf. A minimal foliation means that every leaf is dense; a minimal surface means a surface with vanishing mean curvature for the specified Riemannian metric. These two uses of minimal are distinct. For a cooriented codimension-one foliation choose a nowhere-zero $1$-form $\alpha$ with $T\mathcal F=\ker\alpha$. Integrability gives $d\alpha=\alpha\wedge\omega$. The Godbillon--Vey class is $\operatorname{gv}(\mathcal F)=[\omega\wedge d\omega]\in H^3(M;\mathbb R)$, independent of these choices, and its value on an oriented closed $M$ is the integral of this $3$-form. An ordinary foliated cobordism between foliated $3$-manifolds is a compact $4$-dimensional cobordism with a codimension-one foliation transverse to its boundary and restricting to the given boundary foliations. A taut version requires an additional suitable notion of tautness for this cobordism; choosing and justifying that notion is part of the problem.

Statement recorded in the supplied source.

Formulate a useful notion of taut foliated cobordism and determine numerical invariants of the induced equivalence classes of taut foliations. Seek invariants finer than the Godbillon--Vey number, capable of distinguishing classes that this number alone does not separate. [S2]

Short English statement

Develop taut foliated cobordism and numerical invariants of its classes that distinguish more than the Godbillon--Vey invariant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300067 (AMR-102-0067). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Danny Calegari, Problems in foliations and laminations of 3-manifolds, version 0.78 (2002), Question 15.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.