Left orderability, foliations, and transverse $({\pi }_1,\mathbb{R})$ structures for $3$-manifolds with sphere boundary

Abstract

Let $M$ be a closed orientable irreducible $3$-manifold such that ${\pi }_1(M)$ is left orderable. (a) Let $M_0=M-Int(B^3)$, where $B^3$ is a compact $3$-ball in $M$. We have a process to produce a co-orientable Reebless foliation $\mathcal{F}$ in $M_0$ such that: (1) $\mathcal{F}$ has a transverse $({\pi }_1(M),\mathbb{R})$ structure, (2) there exists a simple closed curve in $M$ that is co-orientably transverse to $\mathcal{F}$ and intersects every leaf of $\mathcal{F}$. More specifically, given a pair $(<,\Gamma )$ composed of a left-invariant order "$<$" of ${\pi }_1(M)$ and a fundamental domain $\Gamma$ of $M$ in its universal cover with certain property (which always exists), we can produce a resulting foliation in $M-Int(B^3)$ as above, and we can test if it can extend to a taut foliation of $M$. (b) Suppose further that $M$ is either atoroidal or a rational homology $3$-sphere. If $M$ admits an $\mathbb{R}$-covered foliation ${\mathcal{F}}_0$, then there is a resulting foliation $\mathcal{F}$ of our process in $M-Int(B^3)$ such that: $\mathcal{F}$ can extend to an $\mathbb{R}$-covered foliation ${\mathcal{F}}_{extend}$ of $M$, and ${\mathcal{F}}_0$ can be recovered from doing a collapsing operation on ${\mathcal{F}}_{extend}$. Here, by a collapsing operation on ${\mathcal{F}}_{extend}$, we mean the following process: (1) choosing an embedded product space $S\times I$ in $M$ for some (possibly non-compact) surface $S$ such that $S\times \{0\},S\times \{1\}$ are leaves of ${\mathcal{F}}_{extend}$ (notice that ${\mathcal{F}}_{extend}{\mid }_{S\times I}$ may not be a product bundle), (2) replacing ${\mathcal{F}}_{extend}{\mid }_{S\times I}$ by a single leaf $S$. (c) We conjecture that there always exists a resulting foliation of our process in $M-Int(B^3)$ which can extend to a taut foliation in $M$.

References in corpus (1)

• Approximating $C^0$-foliations by contact structures

Cited by in corpus (1)

• Foliations from left orders