Geometric Topology

New submissions

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New submissions for Tue, 14 May 24

[1]  arXiv:2405.06748 [pdf, other]

Title: On kernels of homological representations of mapping class groups

Authors: Renaud Detcherry, Jules Martel

Comments: 36p. 15 fig

Subjects: Geometric Topology (math.GT)

We study the kernels of representations of mapping class groups of surfaces on twisted homologies of configuration spaces. We relate them with the kernel of a natural twisted intersection pairing: if the latter kernel is trivial then the representation is faithful. As a main example, we study the representations $\rho_{n}$ of $\mathrm{Mod}(\Sigma_{g,1})$ based on a Heisenberg local system on the $n$ points configuration space of $\Sigma_{g,1}$, introduced by Blanchet--Palmer--Shaukat, and some of their specializations. In the one point configuration case, or when the Heisenberg group is quotiented by an element of its center, we find kernel elements in the twisted intersection form. On the other hand, for $n>2$ configuration points and the full Heisenberg local system, we identify subrepresentations of subgroups of $\mathrm{Mod}(\Sigma_{g,1})$ with Lawrence representations. In particular, we find one of these subgroups, which is isomorphic to a pure braid group on $g$ strands, on which the representations $\rho_n$ are faithful.

[2]  arXiv:2405.06976 [pdf, other]

Title: Kirby-Thompson invariants of distant sums of standard surfaces

Authors: Minami Taniguchi

Comments: 18 pages, 19 figures

Subjects: Geometric Topology (math.GT)

Blair, Campisi, Taylor, and Tomova defined the L-invariant L(F) of a knotted surface F, using pants complexes of trisection surfaces of bridge trisections of F. After that, Aranda, Pongtanapaisan, and Zhang introduced the L*-invariant L*(F) using dual curve complexes instead of pants complexes. In this paper, we determine both of L-invariant and L*-invariant of any finite distant sum of standard surfaces, and this is the first example of knotted surfaces whose bridge numbers and these invariants can be arbitrary large.

[3]  arXiv:2405.07375 [pdf, ps, other]

Title: Lie Superalgebra generalizations of the Jaeger-Kauffman-Saleur Invariant

Authors: Micah Chrisman, Anup Poudel

Comments: 42 pages, 29 figures, comments are welcome

Subjects: Geometric Topology (math.GT)

Jaeger-Kauffman-Saleur (JKS) identified the Alexander polynomial with the $U_q(\mathfrak{gl}(1|1))$ quantum invariant of classical links and extended this to a 2-variable invariant of links in thickened surfaces. Here we generalize this story for every Lie superalgebra of type $\mathfrak{gl}(m|n)$. Following Reshetikhin and Turaev, we first define a virtual $U_q(\mathfrak{gl}(m|n))$ functor for virtual tangles. When $m=n=1$, this recovers the Alexander polynomial of almost classical knots, as defined by Boden-Gaudreau-Harper-Nicas-White. Next, an extended $U_q(\mathfrak{gl}(m|n))$ functor of virtual tangles is obtained by applying the Bar-Natan $Zh$-construction. This is equivalent to the 2-variable JKS-invariant when $m=n=1$, but otherwise our invariants are new whenever $n>0$. In contrast with the classical case, the virtual and extended $U_q(\mathfrak{gl}(m|n))$ functors are not entirely determined by the difference $m-n$. For example, the invariants from $U_q(\mathfrak{gl}(2|0))$ (i.e. the classical Jones polynomial) and $U_q(\mathfrak{gl}(3|1))$ are distinct, as are the extended invariants from $U_q(\mathfrak{gl}(1|1))$ and $U_q(\mathfrak{gl}(2|2))$.
The JKS-invariant was previously shown to be a slice obstruction for virtual links. We present computational evidence that each extended $U_q(\mathfrak{gl}(m|m))$ polynomial is also virtual slice obstructions. Assuming this conjecture holds for just $m=2$, it follows that the virtual knots 6.31445 and 6.62002 are not slice. Both these knots have trivial JKS-invariant, trivial graded genus, trivial Rasmussen invariant, and vanishing extended Milnor invariants up to high order, and hence, no other slice obstructions have previously been found.

[4]  arXiv:2405.07707 [pdf, other]

Title: Stein-fillability and positivity in the mapping class group

Authors: Vitalijs Brejevs, Andy Wand

Comments: 6 pages, 2 figures. Comments are welcome!

Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We construct an infinite family of non-positive open books with once-punctured torus pages that support Stein-fillable contact structures. Combined with a result of Wendl, this allows us to give a complete answer to a long-standing question about the mapping class group of a compact surface with boundary: namely, we conclude that the monoid of monodromies supporting Stein-fillable contact structures is equal to the monoid of positive monodromies if and only if the surface is planar.

[5]  arXiv:2405.07928 [pdf, other]

Title: The Casson-Sullivan invariant for homeomorphisms of 4-manifolds

Authors: Daniel A.P. Galvin

Comments: 40 pages, 1 figure. Comments welcome!

Subjects: Geometric Topology (math.GT)

We investigate the realisability of the Casson-Sullivan invariant for homeomorphisms of smooth $4$-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients. We prove that for all orientable pairs of homeomorphic, smooth $4$-manifolds this invariant can be realised fully after stabilising with a single $S^2\times S^2$. As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected $4$-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on $4$-manifolds which are diffeomorphic but not stably pseudo-isotopic.

Cross-lists for Tue, 14 May 24

[6]  arXiv:2405.06982 (cross-list from math.QA) [pdf, other]

Title: Quantum groups from homologies of configuration spaces

Authors: Stephen Bigelow, Jules Martel

Comments: 46p. many figures

Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT)

We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our diagrams represent true submanifolds of configuration spaces and combinatorial relations between them translate actual twisted homological relations.

[7]  arXiv:2405.07598 (cross-list from math.DG) [pdf, other]

Title: Filling Riemann surfaces by hyperbolic Schottky manifolds of negative volume

Authors: Tommaso Cremaschi, Viola Giovannini, Jean-Marc Schlenker

Subjects: Differential Geometry (math.DG); High Energy Physics - Theory (hep-th); Geometric Topology (math.GT)

We provide conditions under which a Riemann surface $X$ is the asymptotic boundary of a convex co-compact hyperbolic manifold, homeomorphic to a handlebody, of negative renormalized volume. We prove that this is the case when there are on $X$ enough closed curves of short enough hyperbolic length.

[8]  arXiv:2405.07837 (cross-list from q-bio.NC) [pdf, ps, other]

Title: Why Decussate? Topological Constraints on 3D Wiring

Authors: Troy Shinbrot, Wise Young

Comments: 15 pages, 8 figures

Journal-ref: The Anatomical Record 291.10 (2008) 1278-1292

Subjects: Neurons and Cognition (q-bio.NC); Disordered Systems and Neural Networks (cond-mat.dis-nn); Geometric Topology (math.GT)

Many vertebrate motor and sensory systems decussate, or cross the midline to the opposite side of the body. The successful crossing of millions of axons during development requires a complex of tightly controlled regulatory processes. Because these processes have evolved in many distinct systems and organisms, it seems reasonable to presume that decussation confers a significant functional advantage. Yet if this is so, the nature of this advantage is not understood. In this article, we examine constraints imposed by topology on the ways that a three-dimensional processor and environment can be wired together in a continuous, somatotopic, way. We show that as the number of wiring connections grows, decussated arrangements become overwhelmingly more robust against wiring errors than seemingly simpler same-sided wiring schemes. These results provide a predictive approach for understanding how 3D networks must be wired if they are to be robust, and therefore have implications both for future large-scale computational networks and for complex bio-medical devices

[9]  arXiv:2405.07954 (cross-list from math.DS) [pdf, other]

Title: An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions

Authors: Inti Cruz Diaz

Comments: PhD thesis from the University of Burgundy Franche-Comt\'e"

Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

This thesis provides a classification of generalized pseudo-Anosov homeomorphisms up to topological conjugacy using an algorithmic approach. A Markov partition of a generalized pseudo-Anosov homeomorphism is a decomposition of the surface into a finite number of rectangles with disjoint interiors, such that their images intersect with any other rectangle in the Markov partition along a finite number of horizontal sub-rectangles. Every generalized pseudo-Anosov homeomorphism has a Markov partition, and, by using the surface's orientation, we can endow any Markov partition with a geometrization. The geometric type of a geometric Markov partition was defined by Bonatti and Langevin in their book, "Diffeomorphismes de Smale des surfaces", to classify saddle-type basic pieces for structurally stable diffeomorphisms on surfaces. A geometric type is an abstract combinatorial object that generalizes the incidence matrix of a Markov partition. It takes into account not only the number of times the image of a rectangle intersects with any other rectangle in the family but also the order and change of orientation induced by the homeomorphisms. This thesis employs the geometric type of a geometric Markov partition to classify conjugacy classes of pseudo-Anosov homeomorphisms. The classification is provided by the three main results in this manuscript: I) The geometric type is a complete invariant of conjugation. II) A criterion is provided for determining whether an abstract geometric type is realized by a geometric Markov partition of a pseudo-Anosov homeomorphism. III) An algorithm is described for determining whether two geometric types in the pseudo-Anosov class are realized by generalized pseudo-Anosov homeomorphisms that are topologically conjugated or not.

Replacements for Tue, 14 May 24

[10]  arXiv:1812.02295 (replaced) [pdf, ps, other]

Title: On ends of finite-volume noncompact manifolds of nonpositive curvature

Authors: Ran Ji, Yunhui Wu

Comments: Inventiones Mathematicae, to appear, 36 pages

Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

[11]  arXiv:2305.08402 (replaced) [pdf, other]

Title: Adjoint Reidemeister torsions of some 3-manifolds obtained by Dehn surgeries

Authors: Naoko Wakijo

Comments: 14 pages, 1 figure. Comments are welcome. In this revision, we made minor corrections

Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph)

[12]  arXiv:2309.10590 (replaced) [pdf, other]

Title: Boolean algebra on region crossing change and an inequality of the region unknotting number

Authors: Dawan Chumpungam, Ayaka Shimizu

Comments: 13 pages, 14 figures

Subjects: Geometric Topology (math.GT)

[13]  arXiv:2403.15960 (replaced) [pdf, other]

Title: The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

Authors: Benson Farb, Eduard Looijenga

Comments: 31 pages, 4 figures. References to work of Hacking-Keating added to this version

Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Number Theory (math.NT)

[14]  arXiv:2405.04337 (replaced) [pdf, other]

Title: On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$

Authors: Rhea Palak Bakshi, Seongjeong Kim, Xiao Wang

Comments: 30 pages, 20 figures

Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

[15]  arXiv:2303.14503 (replaced) [pdf, ps, other]

Title: On drawing $K_5$ minus an edge in the plane

Authors: T. R. Garaev

Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

[16]  arXiv:2403.18261 (replaced) [pdf, ps, other]

Title: Simplicity of the contactomorphism group of finite regularity

Authors: Yong-Geun Oh

Comments: 61 pages, comments welcome!; v2) many typos corrected, English and exposition improved

Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

[17]  arXiv:2405.00249 (replaced) [pdf, ps, other]

Title: Bi-Lipschitz rigidity of discrete subgroups

Authors: Richard Canary, Hee Oh, Andrew Zimmer

Subjects: Group Theory (math.GR); Dynamical Systems (math.DS); Geometric Topology (math.GT)

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