insight - ScientificComputing - # Virtual Euler Class

A Criterion for Virtual Euler Class One Using Nonvanishing Alexander Polynomials

Q: How might the techniques presented in this paper be extended or modified to address other open problems in 3-manifold topology beyond the virtual Euler class one conjecture?

The techniques presented in the paper, particularly the medley construction and the use of Alexander polynomials as an obstruction criterion, offer potential avenues for exploring other open problems in 3-manifold topology. Here are some possible directions: Virtual properties of other foliation invariants: Instead of focusing on the real Euler class, one could investigate the virtual behavior of other invariants associated with taut foliations, such as the homotopy class of the plane field, the Thurston norm of the Euler class, or the veering structure induced on pseudo-Anosov flows carried by the foliation. The medley construction could be adapted to manipulate these invariants in finite covers. Relationship with Heegaard Floer homology: Heegaard Floer homology is a powerful invariant of 3-manifolds that is closely related to taut foliations. It would be interesting to explore whether the Alexander polynomial criterion or the medley construction can be translated into the Heegaard Floer setting, potentially leading to new insights into the structure of Heegaard Floer homology groups for finite covers. Generalizations to other classes of 3-manifolds: While the paper focuses on hyperbolic 3-manifolds, it is natural to ask whether similar techniques can be applied to other classes, such as Seifert fibered spaces or graph manifolds. This would require understanding the behavior of taut foliations and Alexander polynomials in these settings. Connections with contact topology: Taut foliations are intimately connected to tight contact structures on 3-manifolds. The techniques presented in the paper might shed light on the existence of virtually tight contact structures, i.e., contact structures that become tight after lifting to a finite cover.

Q: Could there be alternative criteria, perhaps not relying on Alexander polynomials, that could provide a complete characterization of when a rational point is realizable as the real Euler class of a taut foliation on some finite cover?

While the paper utilizes Alexander polynomials effectively, it's conceivable that alternative criteria exist for characterizing realizable rational points as real Euler classes in finite covers. Some potential avenues include: Twisted Alexander polynomials and representations: Twisted Alexander polynomials, associated with representations of the fundamental group, offer a richer source of information compared to their classical counterparts. Investigating their behavior under finite covers might provide finer obstructions or even sufficient conditions for realizability. Heegaard Floer homology and related invariants: As mentioned earlier, Heegaard Floer homology and its variants, such as sutured Floer homology and knot Floer homology, are deeply intertwined with taut foliations. Analyzing the behavior of these invariants under finite covers could lead to new criteria for realizability. Geometric techniques and hyperbolic geometry: Exploiting the interplay between taut foliations and the geometry of hyperbolic 3-manifolds might yield geometric criteria. For instance, studying the action of the fundamental group on the circle at infinity of hyperbolic space, or analyzing the veering structure induced by pseudo-Anosov flows carried by the foliation, could provide valuable insights. Obstructions from orderable groups: The fundamental groups of 3-manifolds admitting taut foliations often exhibit interesting orderability properties. Investigating these properties in the context of finite covers might lead to new obstructions for realizability.

Q: What are the implications of the potential connection between the virtual Euler class one conjecture and the study of quantum invariants of 3-manifolds, given the known relationships between taut foliations and certain quantum invariants?

The potential connection between the virtual Euler class one conjecture and quantum invariants is intriguing, given the established relationships between taut foliations and invariants like the Turaev-Viro invariants and the Reshetikhin-Turaev invariants. Here are some possible implications: New insights into quantum invariants: The virtual Euler class one conjecture, if true, could provide new perspectives on the behavior of quantum invariants under finite covers. For instance, it might imply certain stability properties or predictable patterns in the growth of these invariants as one ascends through finite covers. Constraints on taut foliations from quantum invariants: Conversely, the structure and properties of quantum invariants could impose constraints on the possible real Euler classes of taut foliations on finite covers. This could lead to new obstructions or potentially even a proof of the conjecture in specific cases. Connections with topological quantum field theory: The virtual Euler class one conjecture, if framed within the framework of topological quantum field theory (TQFT), might lead to a deeper understanding of the relationship between the geometric structures of taut foliations and the algebraic structures of TQFTs. Categorification and higher representation theory: The study of quantum invariants is closely related to categorification and higher representation theory. The virtual Euler class one conjecture, if it can be formulated in this language, might reveal new connections between the geometry of 3-manifolds and these areas of mathematics.

Core Concepts

The paper presents a criterion for the virtual Euler class one conjecture, which posits that any rational point of dual Thurston norm 1 in the second cohomology group of a closed hyperbolic 3-manifold can be realized as the real Euler class of a taut foliation on some finite cover of the manifold. The criterion hinges on the existence of certain nonvanishing Alexander polynomials associated with the manifold.

Abstract

Bibliographic Information: Liu, Y. (2024). A criterion for virtual Euler class one. arXiv preprint arXiv:2411.11492v1.
Research Objective: This paper aims to provide a criterion for Yazdi's virtual Euler class one conjecture, which deals with the realizability of rational points in the second cohomology group of a closed hyperbolic 3-manifold as real Euler classes of taut foliations on finite covers.
Methodology: The author utilizes techniques from 3-manifold topology, specifically focusing on taut foliations, their Euler classes, and the properties of Alexander polynomials. The core of the proof relies on a novel "medley construction" that allows for the controlled construction of taut foliations on finite cyclic covers of the manifold.
Key Findings: The main result is a criterion stating that for a given rational point in the dual Thurston norm unit ball of a closed hyperbolic 3-manifold, the existence of certain nonvanishing Alexander polynomials guarantees the existence of a finite cyclic cover where the pullback of the given point is realizable as the real Euler class of a taut foliation. This criterion is then applied to construct examples supporting the virtual Euler class one conjecture for manifolds with first Betti numbers 2 or 3 and partial examples for those with first Betti numbers greater than or equal to 4.
Main Conclusions: The paper provides strong evidence for the virtual Euler class one conjecture by establishing a concrete criterion based on Alexander polynomials. The author also suggests that the conjecture might be generalizable to all rational points of dual Thurston norm 1, not just even lattice points. The paper concludes by proposing a "vanishing Alexander polynomial conjecture" as a potential avenue for further research, which could have implications for understanding the growth of the first Betti number in finite covers of 3-manifolds.
Significance: This work contributes significantly to the study of taut foliations and the topology of 3-manifolds. The new criterion offers a powerful tool for constructing taut foliations with prescribed Euler classes, potentially leading to a better understanding of the virtual properties of 3-manifolds.
Limitations and Future Research: The main limitation lies in the partial nature of the results for manifolds with first Betti numbers greater than or equal to 4. The author suggests that further development of virtual construction techniques might be necessary to overcome this limitation and potentially prove the virtual Euler class one conjecture in its entirety. The proposed vanishing Alexander polynomial conjecture offers another promising direction for future research, with potential implications for the virtual Betti number conjecture.

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Quotes

"The proof of our criterion suggests that the even lattice point condition becomes less restrictive in the virtual context. It appears more natural to extend the statement of Conjecture 1.1 to all rational points in H2(M; R) of dual Thurston norm 1."
"A proof of Conjecture 1.6 will imply a positive answer to the question of Cochran and Masters with B = 4."

Key Insights Distilled From

A criterion for virtual Euler class one

by Yi Liu at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11492.pdf

Deeper Inquiries

How might the techniques presented in this paper be extended or modified to address other open problems in 3-manifold topology beyond the virtual Euler class one conjecture?

The techniques presented in the paper, particularly the medley construction and the use of Alexander polynomials as an obstruction criterion, offer potential avenues for exploring other open problems in 3-manifold topology. Here are some possible directions:

Virtual properties of other foliation invariants:  Instead of focusing on the real Euler class, one could investigate the virtual behavior of other invariants associated with taut foliations, such as the homotopy class of the plane field, the Thurston norm of the Euler class, or the veering structure induced on pseudo-Anosov flows carried by the foliation. The medley construction could be adapted to manipulate these invariants in finite covers.

Relationship with Heegaard Floer homology:  Heegaard Floer homology is a powerful invariant of 3-manifolds that is closely related to taut foliations. It would be interesting to explore whether the Alexander polynomial criterion or the medley construction can be translated into the Heegaard Floer setting, potentially leading to new insights into the structure of Heegaard Floer homology groups for finite covers.

Generalizations to other classes of 3-manifolds: While the paper focuses on hyperbolic 3-manifolds, it is natural to ask whether similar techniques can be applied to other classes, such as Seifert fibered spaces or graph manifolds. This would require understanding the behavior of taut foliations and Alexander polynomials in these settings.

Connections with contact topology: Taut foliations are intimately connected to tight contact structures on 3-manifolds. The techniques presented in the paper might shed light on the existence of virtually tight contact structures, i.e., contact structures that become tight after lifting to a finite cover.

Could there be alternative criteria, perhaps not relying on Alexander polynomials, that could provide a complete characterization of when a rational point is realizable as the real Euler class of a taut foliation on some finite cover?

While the paper utilizes Alexander polynomials effectively, it's conceivable that alternative criteria exist for characterizing realizable rational points as real Euler classes in finite covers. Some potential avenues include:

Twisted Alexander polynomials and representations: Twisted Alexander polynomials, associated with representations of the fundamental group, offer a richer source of information compared to their classical counterparts. Investigating their behavior under finite covers might provide finer obstructions or even sufficient conditions for realizability.

Heegaard Floer homology and related invariants: As mentioned earlier, Heegaard Floer homology and its variants, such as sutured Floer homology and knot Floer homology, are deeply intertwined with taut foliations.  Analyzing the behavior of these invariants under finite covers could lead to new criteria for realizability.

Geometric techniques and hyperbolic geometry:  Exploiting the interplay between taut foliations and the geometry of hyperbolic 3-manifolds might yield geometric criteria. For instance, studying the action of the fundamental group on the circle at infinity of hyperbolic space, or analyzing the veering structure induced by pseudo-Anosov flows carried by the foliation, could provide valuable insights.

Obstructions from orderable groups: The fundamental groups of 3-manifolds admitting taut foliations often exhibit interesting orderability properties. Investigating these properties in the context of finite covers might lead to new obstructions for realizability.

What are the implications of the potential connection between the virtual Euler class one conjecture and the study of quantum invariants of 3-manifolds, given the known relationships between taut foliations and certain quantum invariants?

The potential connection between the virtual Euler class one conjecture and quantum invariants is intriguing, given the established relationships between taut foliations and invariants like the Turaev-Viro invariants and the Reshetikhin-Turaev invariants. Here are some possible implications:

New insights into quantum invariants: The virtual Euler class one conjecture, if true, could provide new perspectives on the behavior of quantum invariants under finite covers. For instance, it might imply certain stability properties or predictable patterns in the growth of these invariants as one ascends through finite covers.

Constraints on taut foliations from quantum invariants: Conversely, the structure and properties of quantum invariants could impose constraints on the possible real Euler classes of taut foliations on finite covers. This could lead to new obstructions or potentially even a proof of the conjecture in specific cases.

Connections with topological quantum field theory: The virtual Euler class one conjecture, if framed within the framework of topological quantum field theory (TQFT), might lead to a deeper understanding of the relationship between the geometric structures of taut foliations and the algebraic structures of TQFTs.

Categorification and higher representation theory:  The study of quantum invariants is closely related to categorification and higher representation theory. The virtual Euler class one conjecture, if it can be formulated in this language, might reveal new connections between the geometry of 3-manifolds and these areas of mathematics.