Geometric Topology
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New submissions for Fri, 7 Jun 24
- [1] arXiv:2406.03754 [pdf, other]
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Title: Classification of generalized torsion elements of order two in 3-manifold groupsComments: 19 pages, 6 figuresSubjects: Geometric Topology (math.GT); Group Theory (math.GR)
Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the identity, then $g$ is called a generalized torsion element. The minimum number of conjugates in such a product is called the order of $g$. We will classify $3$-manifolds $M$, each of whose fundamental group has a generalized torsion element of order two. Furthermore, we will classify such elements in $\pi_1(M)$. We also prove that $R$-group and $\overline{R}$-group coincide for $3$-manifold groups, and classify $3$-manifold groups which are $R$-groups (and hence $\overline{R}$-groups).
- [2] arXiv:2406.03800 [pdf, other]
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Title: Constructing embedded surfaces for cellular embeddings of leveled spatial graphsComments: 20 pages, 14 figuresSubjects: Geometric Topology (math.GT); Combinatorics (math.CO)
Finding a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ where a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$ cellular embeds is in general not possible. We therefore restrict our interest to the special class of spatial graphs that are leveled. We show that for leveled spatial graphs with a small number of levels, a surface $\mathcal{S}$ can always be found. The argument is based on the idea of decomposing $\mathcal{G}$ into subgraphs that can be placed on a sphere and on handles that are attached to the sphere, together forming an embedding of $\mathcal{G}$ in $\mathcal{S}$. We generalize the procedure to an algorithm that, if successful, constructs $\mathcal{S}$ for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm.
- [3] arXiv:2406.04293 [pdf, other]
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Title: On the Legendrian realisation of parametric families of knotsAuthors: Javier Martínez-AguinagaComments: 27 pages, 5 figuresSubjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Symplectic Geometry (math.SG)
We study the natural inclusion of the space of Legendrian embeddings in $(\mathbb{S}^3,\xi_{\operatorname{std}})$ into the space of smooth embeddings from a homotopical viewpoint.
T. K\'alm\'an posed in [Kal] the open question of whether for every fixed knot type $\mathcal{K}$ and Legendrian representative $\mathcal{L}$, the homomorphism $\pi_1(\mathcal{L})\to\pi_1(\mathcal{K})$ is surjective. We positively answer this question for infinitely many knot types $\mathcal{K}$ in the three main families (hyperbolic, torus and satellites) and every stabilised Legendrian representative in $(\mathbb{S}^3,\xi_{\operatorname{std}})$.
We then show that for every $n\geq 3$, the homomorphisms $\pi_n(\mathcal{L})\to\pi_n(\mathcal{K})$ and $\pi_n(\mathcal{FL})\to\pi_n(\mathcal{K})$ are never surjective for any knot type $\mathcal K$, Legendrian representative $\mathcal L$ or formal Legendrian representative $\mathcal{FL}$. This shows the existence of rigidity at every higher-homotopy level beyond $\pi_3$. For completeness, we also show that surjectivity at the $\pi_2$-level depends on the smooth knot type. - [4] arXiv:2406.04319 [pdf, other]
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Title: Invariant quasimorphisms and generalized mixed Bavard dualityComments: 57 pages, 11 figuresSubjects: Geometric Topology (math.GT); Group Theory (math.GR)
This article provides an expository account of the celebrated duality theorem of Bavard and three its strengthenings. The Bavard duality theorem connects scl (stable commutator length) and quasimorphisms on a group. Calegari extended the framework from a group element to a chain on the group, and established the generalized Bavard duality. Kawasaki, Kimura, Matsushita and Mimura studied the setting of a pair of a group and its normal subgroup, and obtained the mixed Bavard duality. The first half of the present article is devoted to an introduction to these three Bavard dualities. In the latter half, we present a new strengthening, the generalized mixed Bavard duality, and provide a self-contained proof of it. This third strengthening recovers all of the Bavard dualities treated in the first half; thus, we supply complete proofs of these four Bavard dualities in a unified manner. In addition, we state several results on the space $\mathrm{W}(G,N)$ of non-extendable quasimorphisms, which is related to the comparison problem between scl and mixed scl via the mixed Bavard duality.
Cross-lists for Fri, 7 Jun 24
- [5] arXiv:2406.04258 (cross-list from math.SG) [pdf, other]
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Title: Quiver Hecke algebras from Floer homology in Couloumb branchesComments: 53 pages, 18 figuresSubjects: Symplectic Geometry (math.SG); High Energy Physics - Theory (hep-th); Geometric Topology (math.GT); Representation Theory (math.RT)
Homology theories categorifying quantum group link invariants are known to be governed by the representation theory of quiver Hecke algebras, also called KLRW algebras. Here we show that certain cylindrical KLRW algebras, relevant in particular for cylindrical generalizations of link homology theories, can be realized by Lagrangian Floer homology in multiplicative Coulomb branches. This confirms a homological mirror symmetry prediction of the first author.
Replacements for Fri, 7 Jun 24
- [6] arXiv:1608.06236 (replaced) [pdf, ps, other]
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Title: The homotopy type of the PL cobordism category. IAuthors: Mauricio Gomez LopezComments: This new version incorporates improvements suggested by the referee. 80 pagesSubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
- [7] arXiv:2005.00274 (replaced) [pdf, other]
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Title: Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groupsComments: 17 pages. Minor changes following a referee report. To appear in Mathematical Proceedings of the Cambridge Philosophical SocietySubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
- [8] arXiv:2006.06127 (replaced) [pdf, ps, other]
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Title: Algebraic criteria for stable diffeomorphism of spin 4-manifoldsComments: 102 pages. Version 2: Some results on the Kervaire-Milnor invariant have been extracted to create arXiv:2105.12153. A new Chapter 7 gives an application of our theory. Version 3: Changes following a referee report. Accepted for publication in Memoirs of the American Mathematical SocietySubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
- [9] arXiv:2311.16972 (replaced) [pdf, ps, other]
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Title: A three-manifold invariant from graph configurationsAuthors: Yohan Mandin-HubléComments: 35 pages, 10 figures. Results unchanged. Minor changes (modified title, modified abstract, added section on potential applications, corrected typos)Subjects: Geometric Topology (math.GT)
- [10] arXiv:2401.07269 (replaced) [pdf, other]
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Title: Large alternating Montesinos knots do not admit purely cosmetic surgeriesComments: 22 pages, 20 figuresSubjects: Geometric Topology (math.GT)
- [11] arXiv:2404.07627 (replaced) [pdf, other]
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Title: Simple lift of non-simple closed curvesComments: 22 pages,20 figures. Revised version with minor changes. Comments welcomeSubjects: Geometric Topology (math.GT)
- [12] arXiv:2405.04737 (replaced) [pdf, other]
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Title: Non-orientable 4-genus of torus knotsComments: 13 pages, 2 figures, corrected typo in references, added attribution to Lobb for disproving non-orientable analog of Milnor conjectureSubjects: Geometric Topology (math.GT)
- [13] arXiv:2308.04612 (replaced) [pdf, ps, other]
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Title: The homotopy type of the PL cobordism category. IIAuthors: Mauricio Gomez LopezComments: This new version incorporates improvements suggested by the referee. 50 pagesSubjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
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