math.GT daily digest: 19 submissions for 6 October 2026

The Tuesday, 6 October 2026 arXiv math.GT new-submissions listing contains 19 eligible papers: 10 new submissions and 9 cross submissions. Replacement submissions are excluded.

New submissions

Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds

Authors: Anar Akhmedov, Azer Akhmedov
arXiv: 2610.03756
Comments: 29 pages
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Group Theory (math.GR); Symplectic Geometry (math.SG)

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees in ruled surfaces. With [ p(g-1)=q(h-1)=n, ] the resulting four-manifold has signature zero and Euler characteristic [ e=4n\left(1-\frac1p-\frac1q\right). ] We classify the types having the rational cohomology ring of . They occur only for odd $k$ and, with , , and , are exactly the solutions of [ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. ] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for . Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $k H$ . We prove minimality, boundary incompressibility, an integral Mayer--Vietoris presentation, and a regular cyclic cover of degree . The boundary subgroup embeds in the fundamental group; this gives a nontrivial Bass--Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for and seven for ; for we recover the degree pairs $(2, 3)$ , $(2, 4)$ , and $(3, 3)$ .

Official arXiv abstract page

Integral face-edge duality for knot triangulations

Authors: Sai Kambampati
arXiv: 2610.04103
Comments: 26 pages, 1 figure; exact-arithmetic verification scripts and data included as ancillary files
Subjects: Geometric Topology (math.GT)

For an ordered ideal triangulation of a knot exterior in , we give a cochain construction relating its face matrix $C$ to its preferred-longitude gluing matrices $A, B$ . The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong's generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of and an exact sequence from the two face cokernels to the gluing cokernel, with quotient $Z /2 Z$ . In the unhalved longitude convention this implies . We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen--Kashaev volume conjecture remain open here.

Official arXiv abstract page

Mixed configuration spaces, fixed points and Loop braid groups

Authors: Stavroula Makri
arXiv: 2610.04382
Comments: 17 pages, 1 figure
Subjects: Geometric Topology (math.GT)

We introduce the notion of mixed configuration spaces of the $3$ -ball. That is, we consider both distinct circles and distinct points in the interior of the $3$ -ball and we prove that the projection onto the configuration space of circles of the $3$ -ball is a locally trivial fibration.
The key idea for defining the mixed configuration spaces is to study fixed points of orientation-preserving homeomorphism of the $3$ -ball that leave invariant a trivial link of $n$ components in the interior of the $3$ -ball. In particular, we prove that two fixed points are Nielsen equivalent if and only if the associated loop braids are conjugate by an element of a distinguished free subgroup of rank $n$ . This result stands as a $3$ -dimensional counterpart of the $2$ -dimensional result where fixed points of homeomorphisms of the punctured disc are characterized in terms of braid elements of the classical Artin braid group.

Official arXiv abstract page

The Real Albanese Map and Smooth Topology of the Cartwright--Steger Quotient

Authors: Anar Akhmedov, Sai-Kee Yeung
arXiv: 2610.04460
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

Let $X$ be the Cartwright--Steger surface with its real involution $c$ and Albanese map , and put . Every smooth real Albanese fiber has two ovals and is nondividing. The branch surface satisfies and is nullhomologous modulo two. Every smoothly embedded $+ 1$ -sphere representing the positive generator of meets $B$ in at least six points. Its lift has square two, pairs trivially with , and has no connected embedded representative of genus zero or one. Circle surgery on an Albanese generator is homeomorphic to for either normal framing. Blowing down a positive sphere, if one exists, gives a $Z [Z]$ -homology homotopy equivalent to .

Official arXiv abstract page

Exotic S^2 x S^2 candidates that are just fake

Authors: R. İnanç Baykur
arXiv: 2610.04634
Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We show that two infinite families of homology 's presented in \cite{FPS, AP}, originally proposed as exotic candidates, are aspherical. In particular, none of these manifolds is an exotic . We prove this by reconstructing them from aspherical building blocks, following Edmonds. The same arguments apply to and rule out further candidates for simply connected exotica.

Official arXiv abstract page

Thurston's sphere packing on 3-dimensional manifolds, II

Authors: Xiaokai He, Xu Xu, Chao Zheng
arXiv: 2610.05232
Comments: 29 pages
Subjects: Geometric Topology (math.GT)

In this paper, we study the rigidity of Thurston's hyperbolic sphere packings on 3-dimensional manifolds. We prove that Thurston's hyperbolic sphere packing is locally determined by combinatorial scalar curvature. We further prove the infinitesimal rigidity that Thurston's hyperbolic sphere packing can not be deformed while keeping the combinatorial Ricci curvature fixed. The main tools include a characterization of the admissible space of Thurston's hyperbolic sphere packings, a variational formula for the dihedral angles of a tetrahedron and an identity for the variations of dihedral angles in a tetrahedron generated by Thurston's hyperbolic sphere packings.

Official arXiv abstract page

Lefschetz fibrations with handlebody monodromy

Authors: R. Inanc Baykur, Susumu Hirose
arXiv: 2610.05537
Comments: 6 pages, 1 figure
Subjects: Geometric Topology (math.GT)

We study relatively minimal Lefschetz fibrations over closed oriented surfaces whose monodromy extends over a fixed 3--dimensional handlebody. If $X$ is the total space and $ℓ$ is the number of critical points, we prove that . Such fibrations with exist precisely for fiber genus and base genus , and one can take $ℓ = 1$ for every such pair. We also show that hyperelliptic handlebody monodromy forces the fibration to be a surface bundle and that every meridian twist, separating or nonseparating, has commutator length two in the handlebody group for .

Official arXiv abstract page

Topological Shaking Numbers and Cabling

Authors: Weizhe Shen
arXiv: 2610.06429
Comments: 18 pages
Subjects: Geometric Topology (math.GT)

We give a Seifert-form criterion for upper bounds on topological shaking numbers and prove that isotropic certificates lift under $(m, 1)$ -cabling.

Official arXiv abstract page

Genus-3 Lefschetz pencils on torus bundles

Authors: Evan Huang
arXiv: 2610.06440
Comments: 26 pages, 4 figures
Subjects: Geometric Topology (math.GT)

We use the breeding technique to construct genus-3 Lefschetz pencils for which the total spaces have . We compute the fundamental groups of our Lefschetz pencils and identify them with the fundamental groups of certain torus bundles over tori. Using work of Friedl and Vidussi, the total spaces of our pencils are homeomorphic to torus bundles over tori. These bundles do not admit sections.

Official arXiv abstract page

On Alternateness of the Jones-Kauffman Polynomial of Virtual Links

Authors: Hamid Abchir, Khaled Qazaqzeh, Mohammed Sabak
arXiv: 2610.06752
Comments: 13 pages, 4 Figures, 3 tables
Subjects: Geometric Topology (math.GT)

We study alternateness of the Jones-Kauffman polynomial of virtual links. In particular, we identify a class of virtual links with alternating Jones-Kauffman polynomial that will be called the class of mock-alternating virtual links. Then, we explain how every mock-alternating virtual link can give rise to an infinite family of mock-alternating virtual links using the twisting technique introduced in [4] and later generalized in [5,18]. Finally, we give a connected state expansion of the mock-determinant for checkerboard-colorable links.

Official arXiv abstract page

Cross submissions

Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds

Authors: Jian Ge, Chao Qian
arXiv: 2610.05117
Cross-list: math.DG
Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT); Metric Geometry (math.MG)

We construct smooth collapsing sequences with sectional curvature at least $- 1$ whose limits are topological manifolds with nonzero Kirby--Siebenmann class. For every , the source and limit are simply connected, have dimensions and $n$ , and the metrics have a common positive Ricci lower bound. The limits are homeomorphic to the product of the Chern four-manifold with . We also obtain spin examples and five-dimensional lens-space homotopy types, the latter from finite quotients of one fixed $2$ -connected eight-manifold. Finally, we realize aspherical manifolds with no closed PL representatives of their homotopy types as collapsing limits with scalar curvatures tending uniformly to $+ \infty$ , and establish an obstruction to their approximation with almost nonnegative Ricci curvature.

Official arXiv abstract page

Failure of Ordinary TQFTs to Distinguish Homotopy Type

Authors: Katherine Novey
arXiv: 2610.05480
Cross-list: math.AT
Comments: 42 pages, 1 figure, comments welcome!
Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT); Quantum Algebra (math.QA)

Work by David Reutter and Christopher Schommer-Pries has shown that ordinary smooth TQFTs can distinguish (stable) diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected, closed, smooth 6-manifolds with finite are diffeomorphic if and only if they cannot be distinguished by ordinary TQFTs. The question of whether or not this result holds for all simply connected closed 6-manifolds was open. We consider TQFTs out of the topological and smooth bordism categories, as well as out of a new bordism category constructed in this paper called the formally smooth (FS) bordism category. We present a pair of simply connected closed 6-manifolds with infinite that are not homotopy equivalent, yet are indistinguishable by ordinary TQFTs in all three categories. We extend this counter-example to show that ordinary TQFTs cannot distinguish the homotopy type of simply connected closed manifolds in all dimensions greater than or equal to 6 and non-simply connected closed 5-manifolds. We show that our class of counter-example pairs is spinnable and present a non-spinnable class of counter-examples by introducing a connected summand. Lastly, we use more highly connected analogues of our counter-examples to show that, for all and , there is a pair of -connected closed -manifolds that are not homotopy equivalent, yet cannot be distinguished by ordinary TQFTs.

Official arXiv abstract page

Uniquely Ergodic Branching Subset Currents on Free and Surface Groups

Authors: Ilya Kapovich
arXiv: 2610.05581
Cross-list: math.GR
Comments: 34 pages
Subjects: Group Theory (math.GR); Dynamical Systems (math.DS); Geometric Topology (math.GT)

\emph{Subset currents} on free groups, on surface groups and, more generally, on word-hyperbolic groups, were introduced by Kapovich and Nagnibeda, and further studied by Sasaki. Subset currents extend ordinary geodesic currents and provide measure-theoretic generalizations of conjugacy classes of infinite quasiconvex subgroups. Their supports can be viewed as ``branching laminations".
For free groups and surface groups, we construct non-rational subset currents with genuinely branching supports that carry no other subset currents except scalar multiples. For every nonabelian finite rank free group and every closed hyperbolic surface group, we show that such a current can be chosen so that its support is uncountable, is disjoint from the locus of two-point boundary subsets, and consists entirely of Cantor subsets of the boundary. More generally, we prove that every nonempty metrizable Choquet simplex occurs as a compact convex base for the cone of currents carried by a minimal genuinely branching subset lamination. The same universality phenomenon extends, apart from the usual rational-current terminology, to every non-elementary torsion-free hyperbolic group.
We also study finite-pattern complexity. We prove that if counts the allowed \emph{round patterns} of radius $n$ in a minimal subset lamination $L$ , then forces the carried-current cone to be finite-dimensional, while infinite-dimensionality forces ; this universal threshold is sharp. We show that the coarse growth type of is independent of the chosen free basis and, more generally, of the chosen finite marked graph. We also show that a refined switch system recovers the exact dimension of the carried-current cone.

Official arXiv abstract page

A Complex Structure on : The Genus-One Case of Calabi's Problem

Authors: Min Gao, Zizhou Tang, Wenjiao Yan
arXiv: 2610.06067
Cross-list: math.CV
Comments: 31 pages
Subjects: Complex Variables (math.CV); Geometric Topology (math.GT)

We construct a complex structure on , giving an affirmative answer to the genus-one case of Calabi's problem. The resulting compact complex threefold is non-Kähler and admits no biholomorphic product decomposition.

Official arXiv abstract page

Polynomial filling functions of sublevel sets in $CAT (0)$ cube complexes

Authors: Jannis Weis
arXiv: 2610.06276
Cross-list: math.GR
Comments: 29 pages
Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We give a criterion ensuring that sufficiently high sublevel sets of a discrete Morse function on a $CAT (0)$ cube complex have polynomial integral homological filling functions. As one application, we verify this criterion for the Stein--Farley complexes of the Higman--Thompson groups , , and . Consequently, these groups have polynomial homological filling functions in every degree .

Official arXiv abstract page

Torus Surgery Constraints and Intermediate Covers of the Cartwright--Steger Surface

Authors: Anar Akhmedov, Sai-Kee Yeung
arXiv: 2610.06442
Cross-list: math.SG
Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We study torus surgery on the Cartwright--Steger surface and the degree-two cover construction proposed in \cite[Remark~1]{ASY}. Every torus in this surface is integrally nullhomologous, and surgery on a fixed disjoint collection of such tori cannot decrease the first Betti number. We then determine the nodal-curve subgroup and complete the fundamental-group calculation in the degree-two construction. The resulting minimal symplectic manifold has signature zero and fundamental group $Z /2$ ; for compatible symplectic gluings, its universal cover is an exotic symplectic . The construction on the original Cartwright--Steger surface gives a minimal symplectic manifold with signature $- 1$ and fundamental group .

Official arXiv abstract page

Profinite rigidity and quandle colourings

Authors: Jacques Darn{é}
arXiv: 2610.06485
Cross-list: math.GN
Subjects: General Topology (math.GN); Group Theory (math.GR); Geometric Topology (math.GT)

We show that the numbers of colourings of a link by all finite quandles is equivalent, as an invariant of links, to the profinite completion of the peripheral system. This allows us to reformulate the classical question of the completeness of the former invariant in terms of profinite rigidity. We also develop some profinite quandle theory, and we show that a topological quandle is profinite if and only if it is compact and totally discontinuous and its group of inner automorphism is relatively compact in its homeomorphism group.

Official arXiv abstract page

Symplectic symmetries of a hypothetical exotic : a potential construction or a nonexistence proof

Authors: Weimin Chen
arXiv: 2610.06746
Cross-list: math.SG
Comments: 65 pages, 2 figures, 5 tables
Subjects: Symplectic Geometry (math.SG); Algebraic Geometry (math.AG); Differential Geometry (math.DG); Geometric Topology (math.GT)

This paper is primarily concerned with the existence of a symplectic $4$ -manifold, homeomorphic but not diffeomorphic to , which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic -action for an odd prime $p$ on a rational homology with positive canonical class. Our result suggests that potentially, there are symplectic -actions which are not finite automorphisms of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$ -manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.

Official arXiv abstract page

The Euler characteristic of manifolds

Authors: Renee S. Hoekzema, Luuk Stehouwer, Simona Veselá
arXiv: 2610.06747
Cross-list: math.AT
Comments: Comments welcome!
Subjects: Algebraic Topology (math.AT); Mathematical Physics (math-ph); Geometric Topology (math.GT)

We completely determine the possible parities of the Euler characteristic of closed -manifolds of a given dimension. As an application we partially calculate the controlled cut-and-paste groups , which were previously undetermined.

Official arXiv abstract page

The Tuesday, 6 October 2026 arXiv math.GT new-submissions listing contains 19 eligible papers: 10 new submissions and 9 cross submissions. Replacement submissions are excluded.

New submissions

Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds

Authors: Anar Akhmedov, Azer Akhmedov
arXiv: 2610.03756
Comments: 29 pages
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Group Theory (math.GR); Symplectic Geometry (math.SG)

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p, q \geq 2$ in ruled surfaces. With [ p(g-1)=q(h-1)=n, ] the resulting four-manifold has signature zero and Euler characteristic [ e=4n\left(1-\frac1p-\frac1q\right). ] We classify the types having the rational cohomology ring of $#_{k} (S^{2} \times S^{2})$ . They occur only for odd $k$ and, with $t = (k + 1) /2$ , $a = g - 1$ , and $b = h - 1$ , are exactly the solutions of [ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. ] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for $k \leq 15$ . Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $k H$ . We prove minimality, boundary incompressibility, an integral Mayer--Vietoris presentation, and a regular cyclic cover of degree $lcm (p, q)$ . The boundary subgroup $π_{1} (Σ_{G}) \times Z$ embeds in the fundamental group; this gives a nontrivial Bass--Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for $k = 3$ and seven for $k = 5$ ; for $k = 1$ we recover the degree pairs $(2, 3)$ , $(2, 4)$ , and $(3, 3)$ .

Official arXiv abstract page

Integral face-edge duality for knot triangulations

Authors: Sai Kambampati
arXiv: 2610.04103
Comments: 26 pages, 1 figure; exact-arithmetic verification scripts and data included as ancillary files
Subjects: Geometric Topology (math.GT)

For an ordered ideal triangulation of a knot exterior in $S^{3}$ , we give a cochain construction relating its face matrix $C$ to its preferred-longitude gluing matrices $A, B$ . The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong's generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of $ker_{Z} B$ and an exact sequence from the two face cokernels to the gluing cokernel, with quotient $Z /2 Z$ . In the unhalved longitude convention this implies $∣ det B ∣ = 2∣ det C ∣^{2}$ . We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen--Kashaev volume conjecture remain open here.

Official arXiv abstract page

Mixed configuration spaces, fixed points and Loop braid groups

Authors: Stavroula Makri
arXiv: 2610.04382
Comments: 17 pages, 1 figure
Subjects: Geometric Topology (math.GT)

We introduce the notion of mixed configuration spaces of the $3$ -ball. That is, we consider both distinct circles and distinct points in the interior of the $3$ -ball and we prove that the projection onto the configuration space of circles of the $3$ -ball is a locally trivial fibration.
The key idea for defining the mixed configuration spaces is to study fixed points of orientation-preserving homeomorphism of the $3$ -ball that leave invariant a trivial link of $n$ components in the interior of the $3$ -ball. In particular, we prove that two fixed points are Nielsen equivalent if and only if the associated loop braids are conjugate by an element of a distinguished free subgroup of rank $n$ . This result stands as a $3$ -dimensional counterpart of the $2$ -dimensional result where fixed points of homeomorphisms of the punctured disc are characterized in terms of braid elements of the classical Artin braid group.

Official arXiv abstract page

The Real Albanese Map and Smooth Topology of the Cartwright--Steger Quotient

Authors: Anar Akhmedov, Sai-Kee Yeung
arXiv: 2610.04460
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

Let $X$ be the Cartwright--Steger surface with its real involution $c$ and Albanese map $α : X \to E$ , and put $Y = X / ⟨ c ⟩$ . Every smooth real Albanese fiber has two ovals and is nondividing. The branch surface $B ≅ #_{3} RP^{2}$ satisfies $j_{*} π_{1} (B) = 4 Z \subset π_{1} (Y) ≅ Z$ and is nullhomologous modulo two. Every smoothly embedded $+ 1$ -sphere representing the positive generator of $H_{2} (Y; Z)$ meets $B$ in at least six points. Its lift has square two, pairs trivially with $K_{X}$ , and has no connected embedded representative of genus zero or one. Circle surgery on an Albanese generator is homeomorphic to $CP^{2}$ for either normal framing. Blowing down a positive sphere, if one exists, gives a $Z [Z]$ -homology $S^{1} \times S^{3}$ homotopy equivalent to $S^{1} \times S^{3}$ .

Official arXiv abstract page

Exotic S^2 x S^2 candidates that are just fake

Authors: R. İnanç Baykur
arXiv: 2610.04634
Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We show that two infinite families of homology $S^{2} \times S^{2}$ 's presented in \cite{FPS, AP}, originally proposed as exotic candidates, are aspherical. In particular, none of these manifolds is an exotic $S^{2} \times S^{2}$ . We prove this by reconstructing them from aspherical building blocks, following Edmonds. The same arguments apply to and rule out further candidates for simply connected exotica.

Official arXiv abstract page

Thurston's sphere packing on 3-dimensional manifolds, II

Authors: Xiaokai He, Xu Xu, Chao Zheng
arXiv: 2610.05232
Comments: 29 pages
Subjects: Geometric Topology (math.GT)

In this paper, we study the rigidity of Thurston's hyperbolic sphere packings on 3-dimensional manifolds. We prove that Thurston's hyperbolic sphere packing is locally determined by combinatorial scalar curvature. We further prove the infinitesimal rigidity that Thurston's hyperbolic sphere packing can not be deformed while keeping the combinatorial Ricci curvature fixed. The main tools include a characterization of the admissible space of Thurston's hyperbolic sphere packings, a variational formula for the dihedral angles of a tetrahedron and an identity for the variations of dihedral angles in a tetrahedron generated by Thurston's hyperbolic sphere packings.

Official arXiv abstract page

Lefschetz fibrations with handlebody monodromy

Authors: R. Inanc Baykur, Susumu Hirose
arXiv: 2610.05537
Comments: 6 pages, 1 figure
Subjects: Geometric Topology (math.GT)

We study relatively minimal Lefschetz fibrations over closed oriented surfaces whose monodromy extends over a fixed 3--dimensional handlebody. If $X$ is the total space and $ℓ$ is the number of critical points, we prove that $σ (X) = - ℓ$ . Such fibrations with $ℓ > 0$ exist precisely for fiber genus $g \geq 3$ and base genus $h \geq 2$ , and one can take $ℓ = 1$ for every such pair. We also show that hyperelliptic handlebody monodromy forces the fibration to be a surface bundle and that every meridian twist, separating or nonseparating, has commutator length two in the handlebody group for $g \geq 3$ .

Official arXiv abstract page

Topological Shaking Numbers and Cabling

Authors: Weizhe Shen
arXiv: 2610.06429
Comments: 18 pages
Subjects: Geometric Topology (math.GT)

We give a Seifert-form criterion for upper bounds on topological shaking numbers and prove that isotropic certificates lift under $(m, 1)$ -cabling.

Official arXiv abstract page

Genus-3 Lefschetz pencils on torus bundles

Authors: Evan Huang
arXiv: 2610.06440
Comments: 26 pages, 4 figures
Subjects: Geometric Topology (math.GT)

We use the breeding technique to construct genus-3 Lefschetz pencils for which the total spaces have $c_{1} = 0$ . We compute the fundamental groups of our Lefschetz pencils and identify them with the fundamental groups of certain torus bundles over tori. Using work of Friedl and Vidussi, the total spaces of our pencils are homeomorphic to torus bundles over tori. These bundles do not admit sections.

Official arXiv abstract page

On Alternateness of the Jones-Kauffman Polynomial of Virtual Links

Authors: Hamid Abchir, Khaled Qazaqzeh, Mohammed Sabak
arXiv: 2610.06752
Comments: 13 pages, 4 Figures, 3 tables
Subjects: Geometric Topology (math.GT)

We study alternateness of the Jones-Kauffman polynomial of virtual links. In particular, we identify a class of virtual links with alternating Jones-Kauffman polynomial that will be called the class of mock-alternating virtual links. Then, we explain how every mock-alternating virtual link can give rise to an infinite family of mock-alternating virtual links using the twisting technique introduced in [4] and later generalized in [5,18]. Finally, we give a connected state expansion of the mock-determinant for checkerboard-colorable links.

Official arXiv abstract page

Cross submissions

Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds

Authors: Jian Ge, Chao Qian
arXiv: 2610.05117
Cross-list: math.DG
Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT); Metric Geometry (math.MG)

We construct smooth collapsing sequences with sectional curvature at least $- 1$ whose limits are topological manifolds with nonzero Kirby--Siebenmann class. For every $n \geq 6$ , the source and limit are simply connected, have dimensions $n + 3$ and $n$ , and the metrics have a common positive Ricci lower bound. The limits are homeomorphic to the product of the Chern four-manifold with $S^{n - 4}$ . We also obtain spin examples and five-dimensional lens-space homotopy types, the latter from finite quotients of one fixed $2$ -connected eight-manifold. Finally, we realize aspherical manifolds with no closed PL representatives of their homotopy types as collapsing limits with scalar curvatures tending uniformly to $+ \infty$ , and establish an obstruction to their approximation with almost nonnegative Ricci curvature.

Official arXiv abstract page

Failure of Ordinary TQFTs to Distinguish Homotopy Type

Authors: Katherine Novey
arXiv: 2610.05480
Cross-list: math.AT
Comments: 42 pages, 1 figure, comments welcome!
Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT); Quantum Algebra (math.QA)

Work by David Reutter and Christopher Schommer-Pries has shown that ordinary smooth TQFTs can distinguish (stable) diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected, closed, smooth 6-manifolds with finite $π_{2}$ are diffeomorphic if and only if they cannot be distinguished by ordinary TQFTs. The question of whether or not this result holds for all simply connected closed 6-manifolds was open. We consider TQFTs out of the topological and smooth bordism categories, as well as out of a new bordism category constructed in this paper called the formally smooth (FS) bordism category. We present a pair of simply connected closed 6-manifolds with infinite $π_{2}$ that are not homotopy equivalent, yet are indistinguishable by ordinary TQFTs in all three categories. We extend this counter-example to show that ordinary TQFTs cannot distinguish the homotopy type of simply connected closed manifolds in all dimensions greater than or equal to 6 and non-simply connected closed 5-manifolds. We show that our class of counter-example pairs is spinnable and present a non-spinnable class of counter-examples by introducing a $C P^{2}$ connected summand. Lastly, we use more highly connected analogues of our counter-examples to show that, for all $n \geq 2$ and $m \geq 2 n$ , there is a pair of $(2 n - 1)$ -connected closed $(4 n + m)$ -manifolds that are not homotopy equivalent, yet cannot be distinguished by ordinary TQFTs.

Official arXiv abstract page

Uniquely Ergodic Branching Subset Currents on Free and Surface Groups

Authors: Ilya Kapovich
arXiv: 2610.05581
Cross-list: math.GR
Comments: 34 pages
Subjects: Group Theory (math.GR); Dynamical Systems (math.DS); Geometric Topology (math.GT)

\emph{Subset currents} on free groups, on surface groups and, more generally, on word-hyperbolic groups, were introduced by Kapovich and Nagnibeda, and further studied by Sasaki. Subset currents extend ordinary geodesic currents and provide measure-theoretic generalizations of conjugacy classes of infinite quasiconvex subgroups. Their supports can be viewed as ``branching laminations".
For free groups and surface groups, we construct non-rational subset currents with genuinely branching supports that carry no other subset currents except scalar multiples. For every nonabelian finite rank free group and every closed hyperbolic surface group, we show that such a current can be chosen so that its support is uncountable, is disjoint from the locus of two-point boundary subsets, and consists entirely of Cantor subsets of the boundary. More generally, we prove that every nonempty metrizable Choquet simplex occurs as a compact convex base for the cone of currents carried by a minimal genuinely branching subset lamination. The same universality phenomenon extends, apart from the usual rational-current terminology, to every non-elementary torsion-free hyperbolic group.
We also study finite-pattern complexity. We prove that if $R_{L} (n)$ counts the allowed \emph{round patterns} of radius $n$ in a minimal subset lamination $L$ , then $R_{L} (n) = O (n)$ forces the carried-current cone to be finite-dimensional, while infinite-dimensionality forces $R_{L} (n) / n \to \infty$ ; this universal threshold is sharp. We show that the coarse growth type of $R_{L}$ is independent of the chosen free basis and, more generally, of the chosen finite marked graph. We also show that a refined switch system recovers the exact dimension of the carried-current cone.

Official arXiv abstract page

A Complex Structure on $T^{2} \times S^{4}$ : The Genus-One Case of Calabi's Problem

Authors: Min Gao, Zizhou Tang, Wenjiao Yan
arXiv: 2610.06067
Cross-list: math.CV
Comments: 31 pages
Subjects: Complex Variables (math.CV); Geometric Topology (math.GT)

We construct a complex structure on $T^{2} \times S^{4}$ , giving an affirmative answer to the genus-one case of Calabi's problem. The resulting compact complex threefold is non-Kähler and admits no biholomorphic product decomposition.

Official arXiv abstract page

Polynomial filling functions of sublevel sets in $CAT (0)$ cube complexes

Authors: Jannis Weis
arXiv: 2610.06276
Cross-list: math.GR
Comments: 29 pages
Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We give a criterion ensuring that sufficiently high sublevel sets of a discrete Morse function on a $CAT (0)$ cube complex have polynomial integral homological filling functions. As one application, we verify this criterion for the Stein--Farley complexes of the Higman--Thompson groups $F_{d, r}$ , $T_{d, r}$ , and $V_{d, r}$ . Consequently, these groups have polynomial homological filling functions in every degree $k \geq 2$ .

Official arXiv abstract page

Torus Surgery Constraints and Intermediate Covers of the Cartwright--Steger Surface

Authors: Anar Akhmedov, Sai-Kee Yeung
arXiv: 2610.06442
Cross-list: math.SG
Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We study torus surgery on the Cartwright--Steger surface and the degree-two cover construction proposed in \cite[Remark~1]{ASY}. Every torus in this surface is integrally nullhomologous, and surgery on a fixed disjoint collection of such tori cannot decrease the first Betti number. We then determine the nodal-curve subgroup and complete the fundamental-group calculation in the degree-two construction. The resulting minimal symplectic manifold has signature zero and fundamental group $Z /2$ ; for compatible symplectic gluings, its universal cover is an exotic symplectic $31 CP^{2} #31 \overline{CP}^{2}$ . The construction on the original Cartwright--Steger surface gives a minimal symplectic manifold with signature $- 1$ and fundamental group $(Z /2)^{2}$ .

Official arXiv abstract page

Profinite rigidity and quandle colourings

Authors: Jacques Darn{é}
arXiv: 2610.06485
Cross-list: math.GN
Subjects: General Topology (math.GN); Group Theory (math.GR); Geometric Topology (math.GT)

We show that the numbers of colourings of a link by all finite quandles is equivalent, as an invariant of links, to the profinite completion of the peripheral system. This allows us to reformulate the classical question of the completeness of the former invariant in terms of profinite rigidity. We also develop some profinite quandle theory, and we show that a topological quandle is profinite if and only if it is compact and totally discontinuous and its group of inner automorphism is relatively compact in its homeomorphism group.

Official arXiv abstract page

Symplectic symmetries of a hypothetical exotic $C P^{2}$ : a potential construction or a nonexistence proof

Authors: Weimin Chen
arXiv: 2610.06746
Cross-list: math.SG
Comments: 65 pages, 2 figures, 5 tables
Subjects: Symplectic Geometry (math.SG); Algebraic Geometry (math.AG); Differential Geometry (math.DG); Geometric Topology (math.GT)

This paper is primarily concerned with the existence of a symplectic $4$ -manifold, homeomorphic but not diffeomorphic to $C P^{2}$ , which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic $C P^{2}$ to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic $C P^{2}$ with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic $Z_{p}$ -action for an odd prime $p$ on a rational homology $C P^{2}$ with positive canonical class. Our result suggests that potentially, there are symplectic $Z_{p}$ -actions which are not finite automorphisms of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$ -manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.

Official arXiv abstract page

The Euler characteristic of $P i n^{\pm}$ manifolds

Authors: Renee S. Hoekzema, Luuk Stehouwer, Simona Veselá
arXiv: 2610.06747
Cross-list: math.AT
Comments: Comments welcome!
Subjects: Algebraic Topology (math.AT); Mathematical Physics (math-ph); Geometric Topology (math.GT)

We completely determine the possible parities of the Euler characteristic of closed $P i n^{\pm}$ -manifolds of a given dimension. As an application we partially calculate the controlled cut-and-paste groups $S K K_{2 k + 1}^{P i n^{\pm}}$ , which were previously undetermined.

Official arXiv abstract page

New submissions

Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds

Integral face-edge duality for knot triangulations

Mixed configuration spaces, fixed points and Loop braid groups

The Real Albanese Map and Smooth Topology of the Cartwright--Steger Quotient

Exotic S^2 x S^2 candidates that are just fake

Thurston's sphere packing on 3-dimensional manifolds, II

Lefschetz fibrations with handlebody monodromy

Topological Shaking Numbers and Cabling

Genus-3 Lefschetz pencils on torus bundles

On Alternateness of the Jones-Kauffman Polynomial of Virtual Links

Cross submissions

Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds

Failure of Ordinary TQFTs to Distinguish Homotopy Type

Uniquely Ergodic Branching Subset Currents on Free and Surface Groups

A Complex Structure on : The Genus-One Case of Calabi's Problem

Polynomial filling functions of sublevel sets in CAT(0) cube complexes

Torus Surgery Constraints and Intermediate Covers of the Cartwright--Steger Surface

Profinite rigidity and quandle colourings

Symplectic symmetries of a hypothetical exotic : a potential construction or a nonexistence proof

The Euler characteristic of manifolds

Related content

New submissions

Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds

Integral face-edge duality for knot triangulations

Mixed configuration spaces, fixed points and Loop braid groups

The Real Albanese Map and Smooth Topology of the Cartwright--Steger Quotient

Exotic S^2 x S^2 candidates that are just fake

Thurston's sphere packing on 3-dimensional manifolds, II

Lefschetz fibrations with handlebody monodromy

Topological Shaking Numbers and Cabling

Genus-3 Lefschetz pencils on torus bundles

On Alternateness of the Jones-Kauffman Polynomial of Virtual Links

Cross submissions

Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds

Failure of Ordinary TQFTs to Distinguish Homotopy Type

Uniquely Ergodic Branching Subset Currents on Free and Surface Groups

A Complex Structure on T2×S4: The Genus-One Case of Calabi's Problem

Polynomial filling functions of sublevel sets in CAT(0) cube complexes

Torus Surgery Constraints and Intermediate Covers of the Cartwright--Steger Surface

Profinite rigidity and quandle colourings

Symplectic symmetries of a hypothetical exotic CP2: a potential construction or a nonexistence proof

The Euler characteristic of Pin± manifolds

Related content

Polynomial filling functions of sublevel sets in $CAT (0)$ cube complexes

A Complex Structure on $T^{2} \times S^{4}$ : The Genus-One Case of Calabi's Problem

Polynomial filling functions of sublevel sets in $CAT (0)$ cube complexes

Symplectic symmetries of a hypothetical exotic $C P^{2}$ : a potential construction or a nonexistence proof

The Euler characteristic of $P i n^{\pm}$ manifolds