math.GT daily digest: 17 submissions for 9 October 2026

The Friday, 9 October 2026 arXiv math.GT new-submissions listing contains 17 eligible papers: 13 new submissions and 4 cross submissions. Replacement submissions are excluded.

New submissions

Foliations forcing closed orbits

Authors: Ellis Buckminster, Audrey Rosevear
arXiv: 2610.10558
Comments: 40 pages, 6 figures. Comments welcome!
Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

We study foliations which force every transverse flow to have a closed orbit. We fully characterize this property among finite depth foliations on atoroidal 3-manifolds. We also show that for many foliations, this property can be broken by a -small modification of the foliation. We produce closed orbits using the theory of endperiodic maps, and we construct flows without closed orbits by inserting dynamical plugs transverse to foliations along curves of attracting holonomy.

Official arXiv abstract page

A Faithful Action of the Mapping Class Group on the -Sphere

Authors: Shile Wang
arXiv: 2610.10577
Comments: 12 pages
Subjects: Geometric Topology (math.GT)

We prove that acts faithfully by orientation-preserving homeomorphisms on a -sphere when using equivariant blow-up and bi-orderability of the Torelli group.

Official arXiv abstract page

Finite group actions on nonpositively curved 3-manifolds

Authors: Bena Tshishiku, Zhengyu Zou
arXiv: 2610.10730
Comments: 26 pages
Subjects: Geometric Topology (math.GT)

We show that every orientation-preserving action of a finite group on a closed nonpositively curved 3-manifold preserves a nonpositively curved metric.

Official arXiv abstract page

Bounds on Legendrian Double Twist Knots

Authors: Viktória Földvári, Vera Vértesi
arXiv: 2610.10747
Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We study Legendrian realizations of double twist knots in the standard tight contact 3-sphere. For the family K(4,m), we give upper bounds on the number of oriented and unoriented Legendrian isotopy classes of maximal Thurston--Bennequin representatives. The proof uses bypass techniques in convex surface theory and the classification of tight contact structures.

Official arXiv abstract page

Lifting braids over and non-orientable Lefschetz fibrations

Authors: Valentina Bais, Porter Morgan, Daniele Zuddas
arXiv: 2610.10757
Subjects: Geometric Topology (math.GT)

We show that, given a closed connected non-orientable surface $F$ and a simple branched cover of degree , every self-diffeomorphism of $F$ is isotopic to the lift of a self-diffeomorphism of with respect to $p$ . As an application, we show that every Lefschetz fibration over a connected surface $B$ with and closed non-orientable regular fiber can be expressed as the composition of a simple cover of branched over a surface braided over $B$ and projection onto the $B$ -factor.

Official arXiv abstract page

The rational homology of quotients of the curve complex

Authors: Andrew Putman
arXiv: 2610.10775
Comments: 41 pages, 7 figures
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

Let be the mapping class group of a genus- $g$ surface with $n$ punctures and let be its curve complex. For a finite-index subgroup , Boggi proved that for . This plays an important role in the work of Putman-Wieland on the virtual first Betti number of . The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.

Official arXiv abstract page

Bounded cohomology of transformation groups in all degrees

Authors: Michael Brandenbursky
arXiv: 2610.11327
Comments: 74 pages, 6 figures
Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Symplectic Geometry (math.SG)

Let $M$ be a compact connected oriented smooth manifold, of dimension . Let be one of the following transformation groups: , , (in case $M$ is symplectic) and (in case $M$ is symplectic). Denote by reduced bounded cohomology of in degree $d$ , and by reduced exact bounded cohomology of in degree $d$ . In this paper we prove that if $n = 2$ and $M$ is a closed surface or a disc $D$ , then for every $$\dim(\overline{H}b^d(\mathcal T_M))=\infty.$$ Moreover, if is either or , or ; or . In case , under certain conditions on , we prove that for every degree $$\dim\overline{EH}^d(\mathcal T_M)=\infty.$$ In particular, these results hold for when $M$ is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.

Official arXiv abstract page

A note on realizing triples of slice, concordance and Seifert genera

Authors: Boning Wang
arXiv: 2610.11849
Comments: 5 pages. Comments are welcome
Subjects: Geometric Topology (math.GT)

We show that every triple of integers satisfying is realized as the smooth slice genus, smooth concordance genus, and Seifert genus of a knot, respectively. This proves a conjecture of Kearney.

Official arXiv abstract page

Character variety of the five-punctured sphere and the determinantal quintic hypersurface

Authors: Kazuhiro Hikami
arXiv: 2610.12092
Comments: 16 pages, 6 figures
Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)

Using skein-theoretic methods, we study the SL2(C) character variety of the five-punctured sphere with arbitrary conjugacy class at each of the five punctures. We show that a Fricke-Klein-Vogt-type relation is realized as a symmetric determinantal hypersurface. We also give a realization of the character variety in terms of cluster variables. We explicitly derive the Poisson structure of the simple closed curves on the surface, and prove that cluster mutations induce automorphisms of the character variety.

Official arXiv abstract page

Topological inequalities for conic-line arrangements with some prescribed singularities

Authors: Marco Golla, Piotr Pokora
arXiv: 2610.12192
Comments: 26 pages, no figures. Comments are welcome!
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Combinatorics (math.CO)

We study arrangements of conics and lines in the complex projective plane from the viewpoint of 4-manifolds. We introduce combinatorial conic-line arrangements, the conic-line counterpart of rank-3 matroids, together with their topological and smooth realisations, and we ask which restrictions on a complex arrangement survive when its components are replaced by locally-flat spheres in the same homology classes with the same local singularity models.
For an ADE combinatorial conic-line arrangement of even total degree admitting a topological realisation we prove a Hirzebruch-type inequality; the proof uses only branched covers and the local topology of rational double points. In the complex-algebraic category we show that equality holds exactly when the arrangement is a maximising curve, hence free with prescribed exponents. We then introduce divisible and odd combinatorial conic-line arrangements and obtain inequalities and congruences for the multiplicity sequence of their resolutions, via cyclic branched covers, the G-signature theorem, spin structures, and Furuta's 10/8-theorem.

Official arXiv abstract page

Drilling veering triangulations with applications to pseudo-Anosov flows

Authors: Anna Parlak, Henry Segerman
arXiv: 2610.12302
Comments: 76 pages + appendix with examples (85 pages in total)
Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

We develop an algorithm that takes as input a veering triangulation V and a flow cycle c of V, and outputs a veering parent V_c of V. Constructing veering parents combinatorializes drilling out closed orbits of transitive pseudo-Anosov flows. The algorithm relies on locally approximating the structure of the loom space associated to V in sufficient detail to carry out a certain modification of the Agol-Guéritaud construction.
We use an implementation of the drilling algorithm to construct layered parents of non-layered veering triangulations (and hence Birkhoff sections for transitive pseudo-Anosov flows) and geometric parents of non-geometric veering triangulations. We also find closed orbits of (drilled) pseudo-Anosov flows that are not ambiently isotopic to the geodesics in their free homotopy classes, and show that certain veering triangulations encode almost orbit equivalent flows. In particular, we provide new computational evidence for a conjecture of Ghys asserting that all transitive Anosov flows with orientable invariant foliations are almost orbit equivalent.

Official arXiv abstract page

Coloring graphs of homologous spheres

Authors: Edgar A. Bering IV, Samantha Ford, Aarav Ghai, Maxwell Lyon, Om Shah
arXiv: 2610.12347
Comments: 18 pages, 6 figures
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

The curve graph of a surface is the graph whose vertices are isotopy classes of essential, closed curves on the surface. Edges join vertices with disjoint representatives. The curve graph plays a central role in the theory of mapping class groups of surfaces. Gaster, Greene, and Vlamis have shown that the subgraph of the curve graph induced by curves in a single primitive homology class is uniquely and finitely colorable. In the analogy between the theory of mapping class groups of surfaces and outer automorphisms of free groups, the sphere graph of , a connect sum of $r$ copies of plays an analogous role in the theory. Motivated by Gaster, Green, and Vlamis' result, we investigate , the subgraph of the sphere graph of induced by spheres representing a single primitive homology class . We show that this graph is uniquely colorable. As a corollary, with we obtain a nontrivial homomorphism from the Torelli subgroup to the symmetric group on symbols; this parallels Gaster, Greene, and Vlamis' connection between their coloring and the Chillingworth homomorphism. In the course of our proof we find new smaller generating sets for mapping class groups of with spheres deleted.

Official arXiv abstract page

The Colored Jones Function does not determine any colored Links--Gould polynomial

Authors: Matthew Harper, Ben-Michael Kohli
arXiv: 2610.12352
Comments: 10 pages
Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We prove that there is no uniform description of in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n = 1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita--Terasaka knots by . We develop an odd cabling formula and use a Vandermonde argument to show that for each some satellite of each of these knots is distinguished by .

Official arXiv abstract page

Cross submissions

Braidings of Self-Equivalences and Bordism

Authors: Ian Hambleton, Kursat Sozer
arXiv: 2610.11110
Cross-list: math.AT
Comments: 52 pages
Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

Let $M$ be a closed, smooth or topological $n$ -manifold, with . We construct a homotopy highly cartesian square relating the space $E (M (ℓ))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $ℓ$ -sections of its stable normal microbundle, and an -fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of $E (M (ℓ))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton--Kreck for closed, oriented $4$ -manifolds.

Official arXiv abstract page

Bi-contact plugs and dynamical hyperbolicity

Authors: Surena Hozoori
arXiv: 2610.11243
Cross-list: math.SG
Comments: All comments are welcomed!
Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

We develop a theory of bi-contact plugs in dimension three to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a transversely orientable hyperbolic plug with orientable filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted contact form up to sign. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu under the stated orientability assumptions. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double.
Further applications include a classification of structurally generic transversely oriented projectively Anosov flows without saddle periodic orbits, Morse-Smale examples with prescribed saddle count, a generalization of the construction of Bonatti-Bowden-Potrie to give embeddings of (attracting) hyperbolic plugs into (partially hyperbolic) projectively Anosov flows with the same topological entropy, and constructions on doubles and torus bundles with controlled invariant torus dynamics. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.

Official arXiv abstract page

Self measurable embeddings of mapping class groups

Authors: Alexandra Gurieva
arXiv: 2610.12054
Cross-list: math.GR
Comments: 49 pages, 4 figures. This text is written in Atkinson hyperlegible font. This can be changed to Computer Modern or OpenSans by downloading the sources and commenting/uncommenting the lines as explained in the source tex file
Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

Given a finite-type orientable surface without boundary, outside of a few low-complexity surfaces, we show that any measurable embedding of the associated mapping class group into itself is a measure equivalence. This is a measure-theoretic analog to the result of Ivanov and McCarthy that mapping class groups are co-Hopfian.

Official arXiv abstract page

Higher-rank local mixing for cusped positive representations

Authors: Dongryul M. Kim, Hee Oh, Wenyu Pan
arXiv: 2610.12254
Cross-list: math.DS
Comments: 122 pages
Subjects: Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)

Let be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen--Margulis--Sullivan measures on $ρ (Γ) \ G / M$ , where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup.
For every $u$ in the interior of the limit cone of $ρ (Γ)$ , correlations of smooth compactly supported functions along $exp (t u)$ , normalized by , admit a complete asymptotic expansion in integer powers of . We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on and for orbital counting in Riemannian balls.
More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $Γ$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations.
To prove these results, we develop an intrinsic cusp geometry on the limit set of $ρ (Γ)$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.

Official arXiv abstract page

The Friday, 9 October 2026 arXiv math.GT new-submissions listing contains 17 eligible papers: 13 new submissions and 4 cross submissions. Replacement submissions are excluded.

New submissions

Foliations forcing closed orbits

Authors: Ellis Buckminster, Audrey Rosevear
arXiv: 2610.10558
Comments: 40 pages, 6 figures. Comments welcome!
Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

We study foliations which force every transverse flow to have a closed orbit. We fully characterize this property among $C^{2}$ finite depth foliations on atoroidal 3-manifolds. We also show that for many foliations, this property can be broken by a $C^{0}$ -small modification of the foliation. We produce closed orbits using the theory of endperiodic maps, and we construct $C^{1}$ flows without closed orbits by inserting dynamical plugs transverse to foliations along curves of attracting holonomy.

Official arXiv abstract page

A Faithful Action of the Mapping Class Group $Mod (S_{g})$ on the $(2 g - 1)$ -Sphere

Authors: Shile Wang
arXiv: 2610.10577
Comments: 12 pages
Subjects: Geometric Topology (math.GT)

We prove that $Mod (S_{g})$ acts faithfully by orientation-preserving homeomorphisms on a $(2 g - 1)$ -sphere when $g \geq 3$ using equivariant blow-up and bi-orderability of the Torelli group.

Official arXiv abstract page

Finite group actions on nonpositively curved 3-manifolds

Authors: Bena Tshishiku, Zhengyu Zou
arXiv: 2610.10730
Comments: 26 pages
Subjects: Geometric Topology (math.GT)

We show that every orientation-preserving action of a finite group on a closed nonpositively curved 3-manifold preserves a nonpositively curved metric.

Official arXiv abstract page

Bounds on Legendrian Double Twist Knots

Authors: Viktória Földvári, Vera Vértesi
arXiv: 2610.10747
Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We study Legendrian realizations of double twist knots in the standard tight contact 3-sphere. For the family K(4,m), we give upper bounds on the number of oriented and unoriented Legendrian isotopy classes of maximal Thurston--Bennequin representatives. The proof uses bypass techniques in convex surface theory and the classification of tight contact structures.

Official arXiv abstract page

Lifting braids over $RP^{2}$ and non-orientable Lefschetz fibrations

Authors: Valentina Bais, Porter Morgan, Daniele Zuddas
arXiv: 2610.10757
Subjects: Geometric Topology (math.GT)

We show that, given a closed connected non-orientable surface $F$ and a simple branched cover $p : F \to RP^{2}$ of degree $d \geq 5$ , every self-diffeomorphism of $F$ is isotopic to the lift of a self-diffeomorphism of $RP^{2}$ with respect to $p$ . As an application, we show that every Lefschetz fibration over a connected surface $B$ with $\partial B \neq = \emptyset$ and closed non-orientable regular fiber can be expressed as the composition of a simple cover of $B \times RP^{2}$ branched over a surface braided over $B$ and projection onto the $B$ -factor.

Official arXiv abstract page

The rational homology of quotients of the curve complex

Authors: Andrew Putman
arXiv: 2610.10775
Comments: 41 pages, 7 figures
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

Let $M o d_{g, n}$ be the mapping class group of a genus- $g$ surface with $n$ punctures and let $C_{g, n}$ be its curve complex. For a finite-index subgroup $G < M o d_{g, n}$ , Boggi proved that $H_{k} (C_{g, n} / G; Q) = 0$ for $g + n ≫ k$ . This plays an important role in the work of Putman-Wieland on the virtual first Betti number of $M o d_{g, n}$ . The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.

Official arXiv abstract page

Bounded cohomology of transformation groups in all degrees

Authors: Michael Brandenbursky
arXiv: 2610.11327
Comments: 74 pages, 6 figures
Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Symplectic Geometry (math.SG)

Let $M$ be a compact connected oriented smooth manifold, of dimension $n \geq 2$ . Let $T_{M} G$ be one of the following transformation groups: $H o m e o_{0} (M, μ)$ , $D i f f_{0} (M, μ)$ , $S y m p_{0} (M, ω)$ (in case $M$ is symplectic) and $H am (M, ω)$ (in case $M$ is symplectic). Denote by $\overline{H}_{b}^{d} (T_{M})$ reduced bounded cohomology of $T_{M}$ in degree $d$ , and by $\overline{E H}^{d} (T_{M})$ reduced exact bounded cohomology of $T_{M}$ in degree $d$ . In this paper we prove that if $n = 2$ and $M$ is a closed surface $Σ_{g}$ or a disc $D$ , then for every $d \geq 2$ $$\dim(\overline{H}b^d(\mathcal T_M))=\infty.$$ Moreover, $dim \overline{E H}^{d} (T_{M}) = \infty$ if $T_{M}$ is either $D i f f_{0} (Σ_{g}, μ)$ or $H am (Σ_{g}, ω)$ , or $H am (D, ω)$ ; or $g > 1$ . In case $n > 2$ , under certain conditions on $π_{1} (M)$ , we prove that for every degree $d \geq 2$ $$\dim\overline{EH}^d(\mathcal T_M)=\infty.$$ In particular, these results hold for $T_{M}$ when $M$ is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.

Official arXiv abstract page

A note on realizing triples of slice, concordance and Seifert genera

Authors: Boning Wang
arXiv: 2610.11849
Comments: 5 pages. Comments are welcome
Subjects: Geometric Topology (math.GT)

We show that every triple of integers $(a, b, c)$ satisfying $1 \leq a \leq b \leq c$ is realized as the smooth slice genus, smooth concordance genus, and Seifert genus of a knot, respectively. This proves a conjecture of Kearney.

Official arXiv abstract page

Character variety of the five-punctured sphere and the determinantal quintic hypersurface

Authors: Kazuhiro Hikami
arXiv: 2610.12092
Comments: 16 pages, 6 figures
Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)

Using skein-theoretic methods, we study the SL2(C) character variety of the five-punctured sphere with arbitrary conjugacy class at each of the five punctures. We show that a Fricke-Klein-Vogt-type relation is realized as a symmetric determinantal hypersurface. We also give a realization of the character variety in terms of cluster variables. We explicitly derive the Poisson structure of the simple closed curves on the surface, and prove that cluster mutations induce automorphisms of the character variety.

Official arXiv abstract page

Topological inequalities for conic-line arrangements with some prescribed singularities

Authors: Marco Golla, Piotr Pokora
arXiv: 2610.12192
Comments: 26 pages, no figures. Comments are welcome!
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Combinatorics (math.CO)

We study arrangements of conics and lines in the complex projective plane from the viewpoint of 4-manifolds. We introduce combinatorial conic-line arrangements, the conic-line counterpart of rank-3 matroids, together with their topological and smooth realisations, and we ask which restrictions on a complex arrangement survive when its components are replaced by locally-flat spheres in the same homology classes with the same local singularity models.
For an ADE combinatorial conic-line arrangement of even total degree admitting a topological realisation we prove a Hirzebruch-type inequality; the proof uses only branched covers and the local topology of rational double points. In the complex-algebraic category we show that equality holds exactly when the arrangement is a maximising curve, hence free with prescribed exponents. We then introduce divisible and odd combinatorial conic-line arrangements and obtain inequalities and congruences for the multiplicity sequence of their resolutions, via cyclic branched covers, the G-signature theorem, spin structures, and Furuta's 10/8-theorem.

Official arXiv abstract page

Drilling veering triangulations with applications to pseudo-Anosov flows

Authors: Anna Parlak, Henry Segerman
arXiv: 2610.12302
Comments: 76 pages + appendix with examples (85 pages in total)
Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

We develop an algorithm that takes as input a veering triangulation V and a flow cycle c of V, and outputs a veering parent V_c of V. Constructing veering parents combinatorializes drilling out closed orbits of transitive pseudo-Anosov flows. The algorithm relies on locally approximating the structure of the loom space associated to V in sufficient detail to carry out a certain modification of the Agol-Guéritaud construction.
We use an implementation of the drilling algorithm to construct layered parents of non-layered veering triangulations (and hence Birkhoff sections for transitive pseudo-Anosov flows) and geometric parents of non-geometric veering triangulations. We also find closed orbits of (drilled) pseudo-Anosov flows that are not ambiently isotopic to the geodesics in their free homotopy classes, and show that certain veering triangulations encode almost orbit equivalent flows. In particular, we provide new computational evidence for a conjecture of Ghys asserting that all transitive Anosov flows with orientable invariant foliations are almost orbit equivalent.

Official arXiv abstract page

Coloring graphs of homologous spheres

Authors: Edgar A. Bering IV, Samantha Ford, Aarav Ghai, Maxwell Lyon, Om Shah
arXiv: 2610.12347
Comments: 18 pages, 6 figures
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

The curve graph of a surface is the graph whose vertices are isotopy classes of essential, closed curves on the surface. Edges join vertices with disjoint representatives. The curve graph plays a central role in the theory of mapping class groups of surfaces. Gaster, Greene, and Vlamis have shown that the subgraph of the curve graph induced by curves in a single primitive homology class is uniquely and finitely colorable. In the analogy between the theory of mapping class groups of surfaces and outer automorphisms of free groups, the sphere graph of $M_{r}$ , a connect sum of $r$ copies of $S_{1} \times S_{2}$ plays an analogous role in the theory. Motivated by Gaster, Green, and Vlamis' result, we investigate $S_{v} (M_{r})$ , the subgraph of the sphere graph of $M_{r}$ induced by spheres representing a single primitive homology class $[v] \in H_{2} (M_{r}; Z)$ . We show that this graph is uniquely $r - 1$ colorable. As a corollary, with $r \geq 3$ we obtain a nontrivial homomorphism from the Torelli subgroup $I O_{r} \leq Out (F_{r})$ to the symmetric group on $r - 1$ symbols; this parallels Gaster, Greene, and Vlamis' connection between their coloring and the Chillingworth homomorphism. In the course of our proof we find new smaller generating sets for mapping class groups of $M_{r}$ with spheres deleted.

Official arXiv abstract page

The Colored Jones Function does not determine any colored Links--Gould polynomial

Authors: Matthew Harper, Ben-Michael Kohli
arXiv: 2610.12352
Comments: 10 pages
Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We prove that there is no uniform description of $LG^{(n)}$ in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n = 1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita--Terasaka knots by $LG^{(2)}$ . We develop an odd cabling formula and use a Vandermonde argument to show that for each $n \geq 2$ some satellite of each of these knots is distinguished by $LG^{(n)}$ .

Official arXiv abstract page

Cross submissions

Braidings of Self-Equivalences and Bordism

Authors: Ian Hambleton, Kursat Sozer
arXiv: 2610.11110
Cross-list: math.AT
Comments: 52 pages
Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

Let $M$ be a closed, smooth or topological $n$ -manifold, with $n \geq 4$ . We construct a homotopy highly cartesian square relating the space $E (M (ℓ))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $ℓ$ -sections of its stable normal microbundle, and an $(\infty + n)$ -fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of $E (M (ℓ))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton--Kreck for closed, oriented $4$ -manifolds.

Official arXiv abstract page

Bi-contact plugs and dynamical hyperbolicity

Authors: Surena Hozoori
arXiv: 2610.11243
Cross-list: math.SG
Comments: All comments are welcomed!
Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

We develop a theory of bi-contact plugs in dimension three to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a transversely orientable hyperbolic plug with orientable filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted contact form up to sign. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu under the stated orientability assumptions. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double.
Further applications include a classification of structurally generic transversely oriented projectively Anosov flows without saddle periodic orbits, Morse-Smale examples with prescribed saddle count, a generalization of the construction of Bonatti-Bowden-Potrie to give embeddings of (attracting) hyperbolic plugs into (partially hyperbolic) projectively Anosov flows with the same topological entropy, and constructions on doubles and torus bundles with controlled invariant torus dynamics. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.

Official arXiv abstract page

Self measurable embeddings of mapping class groups

Authors: Alexandra Gurieva
arXiv: 2610.12054
Cross-list: math.GR
Comments: 49 pages, 4 figures. This text is written in Atkinson hyperlegible font. This can be changed to Computer Modern or OpenSans by downloading the sources and commenting/uncommenting the lines as explained in the source tex file
Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

Given a finite-type orientable surface without boundary, outside of a few low-complexity surfaces, we show that any measurable embedding of the associated mapping class group into itself is a measure equivalence. This is a measure-theoretic analog to the result of Ivanov and McCarthy that mapping class groups are co-Hopfian.

Official arXiv abstract page

Higher-rank local mixing for cusped positive representations

Authors: Dongryul M. Kim, Hee Oh, Wenyu Pan
arXiv: 2610.12254
Cross-list: math.DS
Comments: 122 pages
Subjects: Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)

Let $Γ < PSL_{2} (R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $ρ : Γ \to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen--Margulis--Sullivan measures on $ρ (Γ) \ G / M$ , where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup.
For every $u$ in the interior of the limit cone of $ρ (Γ)$ , correlations of smooth compactly supported functions along $exp (t u)$ , normalized by $t^{(rank_{R} G - 1) /2}$ , admit a complete asymptotic expansion in integer powers of $t^{- 1}$ . We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on $L^{2} (ρ (Γ) \ G)$ and for orbital counting in Riemannian balls.
More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $Γ$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations.
To prove these results, we develop an intrinsic cusp geometry on the limit set of $ρ (Γ)$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.

Official arXiv abstract page

New submissions

Foliations forcing closed orbits

A Faithful Action of the Mapping Class Group on the -Sphere

Finite group actions on nonpositively curved 3-manifolds

Bounds on Legendrian Double Twist Knots

Lifting braids over and non-orientable Lefschetz fibrations

The rational homology of quotients of the curve complex

Bounded cohomology of transformation groups in all degrees

A note on realizing triples of slice, concordance and Seifert genera

Character variety of the five-punctured sphere and the determinantal quintic hypersurface

Topological inequalities for conic-line arrangements with some prescribed singularities

Drilling veering triangulations with applications to pseudo-Anosov flows

Coloring graphs of homologous spheres

The Colored Jones Function does not determine any colored Links--Gould polynomial

Cross submissions

Braidings of Self-Equivalences and Bordism

Bi-contact plugs and dynamical hyperbolicity

Self measurable embeddings of mapping class groups

Higher-rank local mixing for cusped positive representations

Related content

New submissions

Foliations forcing closed orbits

A Faithful Action of the Mapping Class Group Mod(Sg​) on the (2g−1)-Sphere

Finite group actions on nonpositively curved 3-manifolds

Bounds on Legendrian Double Twist Knots

Lifting braids over RP2 and non-orientable Lefschetz fibrations

The rational homology of quotients of the curve complex

Bounded cohomology of transformation groups in all degrees

A note on realizing triples of slice, concordance and Seifert genera

Character variety of the five-punctured sphere and the determinantal quintic hypersurface

Topological inequalities for conic-line arrangements with some prescribed singularities

Drilling veering triangulations with applications to pseudo-Anosov flows

Coloring graphs of homologous spheres

The Colored Jones Function does not determine any colored Links--Gould polynomial

Cross submissions

Braidings of Self-Equivalences and Bordism

Bi-contact plugs and dynamical hyperbolicity

Self measurable embeddings of mapping class groups

Higher-rank local mixing for cusped positive representations

Related content

A Faithful Action of the Mapping Class Group $Mod (S_{g})$ on the $(2 g - 1)$ -Sphere

Lifting braids over $RP^{2}$ and non-orientable Lefschetz fibrations