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Affine Hirsch foliations on 3-manifolds

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Bin Yu

Geometric Topology · Submitted 13 Apr 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$-manifold $M$ admits $0$, $1$ or $2$ affine Hirsch foliations.

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arXiv ID1604.03723 (version 3)
Pith IDYP6WFNUD, Pith's permanent record of this paper and its review
CitationBin Yu. "Affine Hirsch foliations on 3-manifolds." arXiv:1604.03723 (2016). https://pith.science/paper/YP6WFNUD
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,596 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

This paper is devoted to discussing affine Hirsch foliations on $3$-manifolds. First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$-manifold $M$ admits $0$, $1$ or $2$ affine Hirsch foliations. Furthermore, every case is possible. Then, we analyze the $3$-manifolds admitting two affine Hirsch foliations (abbreviated as Hirsch manifolds). On the one hand, we construct Hirsch manifolds by using exchangeable braided links (abbreviated as DEBL Hirsch manifolds); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold. Finally, we show that for every $n\in \mathbb{N}$, there are only finitely many Hirsch manifolds with strand number $n$. Here the strand number of a Hirsch manifold $M$ is a positive integer defined by using strand numbers of braids.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 This paper is devoted to discussing affine Hirsch foliations on $3$-manifolds. background 0.53 0.48 0.03 asserted 0.75
s1 First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$-manifold $M$ admits $0$, $1$ or $2$ affine Hirsch foliations. result 0.98 0.93 0.46 asserted 1.00
s2 Furthermore, every case is possible. result 1.00 0.81 0.11 asserted 0.94
s3 Then, we analyze the $3$-manifolds admitting two affine Hirsch foliations (abbreviated as Hirsch manifolds). contribution 0.98 0.85 0.06 asserted 0.81
s4 On the one hand, we construct Hirsch manifolds by using exchangeable braided links (abbreviated as DEBL Hirsch manifolds); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold. result 0.58 0.95 0.26 asserted 1.00
s5 Finally, we show that for every $n\in \mathbb{N}$, there are only finitely many Hirsch manifolds with strand number $n$. result 1.00 0.94 0.27 asserted 1.00
s6 Here the strand number of a Hirsch manifold $M$ is a positive integer defined by using strand numbers of braids. background 0.55 0.62 0.04 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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not examined
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  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

This paper is devoted to discussing affine Hirsch foliations on $3$-manifolds. First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$-manifold $M$ admits $0$, $1$ or $2$ affine Hirsch foliations. Furthermore, every case is possible. Then, we analyze the $3$-manifolds admitting two affine Hirsch foliations (abbreviated as Hirsch manifolds). On the one hand, we construct Hirsch manifolds by using exchangeable braided links (abbreviated as DEBL Hirsch manifolds); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold. Finally, we show that for every $n\in \mathbb{N}$, there are only finitely many Hirsch manifolds with strand number $n$. Here the strand number of a Hirsch manifold $M$ is a positive integer defined by using strand numbers of braids.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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