math.GT daily digest: 11 submissions for 26 October 2026

New submissions

Sphere decomposition for finite subgroups of 3-manifold mapping class groups

• Authors: Corey Bregman, Bena Tshishiku
• arXiv: 2610.31798
• Comments: 25 pages
• Subjects: Geometric Topology (math.GT)

For a closed, oriented 3-manifold $M$, we show that any finite subgroup of its mapping class group preserves an isotopy class of filling sphere system on $M$. This is an isotopy version of the "equivariant sphere theorem" of Meeks-Yau. As part of our analysis, we establish a bijection between simplices of the Guirardel--Levitt outer space for and filling sphere systems in $M$, generalizing an argument of Hatcher. We also compute the image of for any such $M$.

The $d$-invariant of surgeries on $2$-component L-space links

• Authors: Nathan Li
• arXiv: 2610.31853
• Subjects: Geometric Topology (math.GT)

Gorsky, B. Liu and Moore provided a formula for the $d$-invariants of integral surgeries on $2$-component $L$-space links with linking number $0$. This paper generalizes these results to the nonzero linking number case. We describe how the $d$-invariant depends on the definiteness of the framing matrix and provide formulae for certain families of positive definite framings, as well as a skein relation for non-algebraically split links.

Singular Loci of SL2(C)-Representation Varieties of Torus Knot Groups

• Authors: S. Liriano
• arXiv: 2610.32064
• Comments: 8 pages, 1 figure
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Group Theory (math.GR); Representation Theory (math.RT)

Let , where are relatively prime, and let [ R=\Hom(T(q,p),\mathrm{SL}_2(\mathbb C)). ] We compute the integral homology of the singular locus of $R$. We show that this locus has complex dimension $2$ and is a disjoint union of smooth affine surfaces, each homotopy equivalent to the two-dimensional sphere. Consequently, [ H_0(\Sing R;\Z)\cong H_2(\Sing R;\Z)\cong\Z^\mu, ] and all remaining homology groups vanish. Thus both nonzero Betti numbers equal the first Betti number of the Milnor fiber of the plane curve singularity , namely its Milnor number $\mu$.

Conway polynomials of and : a proof of Takemura's conjecture

• Authors: Werner Alois Stanggassinger
• arXiv: 2610.32269
• Comments: 7 pages. Scripts and outputs of the checks: this https URL
• Subjects: Geometric Topology (math.GT)

Takemura conjectured that the Conway polynomials of the Turk's head link and of satisfy for and for , and proved it for . We prove it for all , knots and links. Equivalently, with $c=1$ if is odd and if it is even. The proof combines a product formula , which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid.

Angle cocycles, Euler characteristic, and simplicial volume of smooth manifolds

• Authors: Michelle Bucher, Stefano Francaviglia, Grazia Rago
• arXiv: 2610.32342
• Comments: 37 pages. Comments are welcome!
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

A celebrated conjecture of Gromov states that for closed aspherical manifolds, the vanishing of simplicial volume implies the vanishing of the Euler characteristic. We prove Gromov's conjecture for closed convex projective manifolds and, more generally, for closed orientable smooth manifolds admitting an immersed straightening. Generalising the straightening of simplices in properly convex domains in real projective space, we introduce the notion of immersed straightening and prove that, for every integer , there exists a constant such that every closed orientable smooth $d$-manifold $M$ admitting an immersed straightening satisfies .

Exact bounded cohomology of measure-preserving homeomorphism groups of nonorientable hyperbolic manifolds

• Authors: Michael Brandenbursky
• arXiv: 2610.32475
• Comments: 20 pages
• Subjects: Geometric Topology (math.GT)

Let $M$ be a closed, connected, non-orientable hyperbolic manifold, and let $\mu$ be a good measure on $M$. Denote by the identity component of the group of $\mu$-preserving homeomorphisms. For every , we prove that , where denotes reduced exact bounded cohomology in degree $d$.

Simple homotopy types of smooth four-manifolds with finite fundamental group

• Authors: Daniel Kasprowski, John Nicholson
• arXiv: 2610.32706
• Comments: 8 pages
• Subjects: Geometric Topology (math.GT)

We construct closed smooth $4$-manifolds with finite fundamental group which are homotopy equivalent but not simple homotopy equivalent, even after taking a connected sum with copies of . Our explicit examples have remarkably large fundamental groups; in particular, the smallest examples we obtain have fundamental groups of order roughly .

Counting curves with bounded intersection

• Authors: Hugo Parlier
• arXiv: 2610.32723
• Comments: 29 pages, 8 figures
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

This paper counts curves, and multicurves, on hyperbolic surfaces, providing explicit upper bounds that depend polynomially on length and exponentially in terms of an upper bound on self-intersection.

Veering triangulations of once-punctured torus bundles are unique

• Authors: Saul Schleimer, Henry Segerman
• arXiv: 2610.32771
• Comments: 39 pages, 18 figures and subfigures
• Subjects: Geometric Topology (math.GT)

We show that each hyperbolic once-punctured torus bundle has exactly one veering triangulation. We also find manifolds for which the exact number of veering structures is known and is greater than one. Finally, we give new examples of manifolds without veering triangulations.

Cross submissions

The space of equators has infinite Hofer's diameter

• Authors: Leonid Polterovich, Egor Shelukhin
• arXiv: 2610.31940
• Cross-list: math.SG
• Comments: 26 pages
• Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

We settle several questions about Hofer's geometry of the two-sphere. We prove that the space of equators has infinite Hofer diameter, construct simple commutators of arbitrarily large Hofer norm, and exhibit a Hamiltonian diffeomorphism whose iterates escape to infinity in Hofer distance from the set of autonomous diffeomorphisms. Finally, we construct a bi-Lipschitz group monomorphism from the free group of rank two, equipped with its conjugation-invariant generator norm, into the Hamiltonian group equipped with Hofer's metric. The proof rests on an interplay between Lagrangian Floer theory in the fourth symmetric product of the sphere and combinatorial group theory. This paper was written with substantial assistance from AI.

The elastic figure-eight knot is not planar

• Authors: Tatsuya Miura
• arXiv: 2610.32320
• Cross-list: math.DG
• Comments: 40 pages, 17 figures
• Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Geometric Topology (math.GT)

We prove that every elastic knot in any non-torus twist knot class is not planar. In particular, this disproves a numerically motivated planarity conjecture for elastic figure-eight knots.