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Register a new userLeft orderability, foliations, and transverse $(pi_1,mathbb{R})$ structures for $3$-manifolds with sphere boundary
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Bojun Zhao
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Let $M$ be a closed orientable irreducible $3$-manifold with a left orderable fundamental group, and $M_0 = M - Int(B^{3})$. We show that there exists a Reebless co-orientable foliation $mathcal{F}$ in $M_0$, whose leaves may be transverse to $partial M_0$ or tangent to $partial M_0$ at their intersections with $partial M_0$, such that $mathcal{F}$ has a transverse $(pi_1(M_0),mathbb{R})$ structure, and $mathcal{F}$ is analogue to taut foliations (in closed $3$-manifolds) in the following sense: there exists a compact $1$-manifold (i.e. a finite union of properly embedded arcs and/or simple closed curves) transverse to $mathcal{F}$ that intersects every leaf of $mathcal{F}$. We conjecture that $mathcal{F}$ is obtained from removing a $3$-ball foliated with horizontal disks from a taut foliation in $M$.
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In this paper, we prove that any closed orientable 3-manifold $M$ other than $#^k S^1times S^2$ and $S^3$ satisfies the following properties: (1) For any compact orientable 4-manifold $N$ bounded by $M$, the inclusion does not induce an isomorphism on their fundamental groups $pi_1$. (2) For any map $f:Mto N$ from $M$ to a closed orientable 4-manifold $N$, $f$ does not induce an isomorphism on $pi_1$. Relevant results on higher dimensional manifolds are also obtained.
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Let $N$ be a compact manifold with a foliation $mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,mathscr{F}_N) rightarrow (M,mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,mathscr{F}_N)$ and $h(M,mathscr{F}_M)$ and it holds $h(M,mathscr{F}_M) leq h(N,mathscr{F}_N)$. Additionally, if equality holds, then the leaves must be homothetic.
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