math.GT daily digest: 9 submissions for 5 October 2026

New submissions

Small Lefschetz fibrations on the BMY line

• Authors: R. İnanç Baykur, András I. Stipsicz, Zoltán Szabó
• arXiv: 2610.02261
• Comments: 41 pages
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Symplectic Geometry (math.SG)

We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface. All these examples have only three nodes, and their total spaces are symplectic 4-manifolds of general type on the Bogomolov-Miyaoka-Yau line. In the course of deriving the monodromy relation in genus 19, we obtain several interesting identities, including a factorization of a Dehn twist into a product of three order-3 mapping classes. We show that the total space of this fibration is topologically s-cobordant to the Cartwright-Steger surface, and we determine its real quotient up to homeomorphism. The genus-4 family consists of variations of the Bauer-Catanese fibrations and contains 65 isomorphism classes of Lefschetz fibrations on 45 diffeomorphism types of compact complex ball quotients. We describe their topology and that of the real quotients of the three original surfaces. The handle monodromies in our main factorizations are pseudo-Anosov, and we show more generally that for holomorphic genus-4 Lefschetz fibrations on the BMY line, this holds for every choice of handle generators.

Reflections on branched covers

• Authors: Grigori Avramidi, Boris Okun, Kevin Schreve
• arXiv: 2610.02723
• Comments: 21 pages, 4 figures
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

A flag complex $L$ determines a locally CAT(0) cube complex with the links of all vertices isomorphic to $L$. The fundamental group of is the commutator subgroup of the right-angled Coxeter group . We observe that an edge contraction determines a $2$-fold branched cover (up to a homotopy equivalence). We use this branched cover to give an inequality between the --Betti numbers of and . This reduces an -version of the Singer Conjecture for right-angled Coxeter groups to minimal flag triangulations of even dimensional spheres. We also use this inequality to compute the -Betti numbers of when $L$ is a flag triangulation of the projective plane or the torus.

Quantitative chain extensions and integral cohomological fillings

• Authors: Franziska Hofmann, Clara Loeh
• arXiv: 2610.02987
• Comments: 43 pages
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

We develop two methods in quantitative homological algebra: A declarative approach to integral cohomological filling problems and a quantitatively controlled replacement of normed chain complexes by their homology complex. Both methods are connected through so-called chain extensions. As applications, we obtain transparent proofs for uniform integral simplicial cohomological fillings for subdivisions of simplicial complexes, for uniform integral cohomological fillings of certain finite coverings, and for lower bounds of the -semi-norm on product spaces.

Large-scale rank and geometry of the Lipschitz metric on geodesic currents

• Authors: Dídac Martínez-Granado, Jenya Sapir
• arXiv: 2610.03359
• Comments: 70 pages, 3 figures
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR)

For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by . We then show a sufficient criterion for its rank and asymptotic dimension to be equal to and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to . In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.

Cross submissions

Homotopy Self-Equivalences of Manifolds

• Authors: Mehmetcik Pamuk
• arXiv: 2610.02244
• Cross-list: math.AT
• Comments: 34 pages
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

Let $M$ be a closed, oriented manifold. The group of based self-equivalences of $M$ is a natural invariant in manifold classification, and Hambleton and Kreck's commutative braid gives a systematic way to compute it for $4$-manifolds. We extend their framework to -manifolds of dimension $2n$ or , under two hypotheses: (F), that the structure map factors through the Postnikov $n$-type, and (T), triviality of the relevant monodromy. Both hold whenever $M$ is stably parallelizable, which is the case in every example we treat. We then discuss products and connected sums of spheres. For , the isometry group of the relevant intersection form is the symplectic group when $n$ is odd, and the finite group when $n$ is even, which we verify directly with a Whitehead-product argument. Feeding this into the braid gives, in each case, a short exact sequence , with $Q$ this same isometry group and $K$ a bordism-theoretic kernel built from unstable homotopy groups of spheres. We work this out for ; for , where the sequence splits as a direct product; for connected sums of $r$ copies of , where permuting the summands acts non-trivially on $K$ instead; for ; and for , where $Q$ is fully determined but $K$ is only conjectured. We also discuss the connected sum \connsum , where the same method gets most of the way to an answer without quite closing it. Throughout, comparing with the smooth structure set shows how differently homotopy and diffeomorphism classification can behave.

Ricci flow on manifolds with positive isotropic curvature in all dimensions

• Authors: Simon Brendle, Raphael Tsiamis
• arXiv: 2610.02325
• Cross-list: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We extend the classification of manifolds with positive isotropic curvature to all dimensions . This result also completes the classification of uniformly positive isotropic curvature on complete non-compact $\kappa$-noncollapsed ancient solutions and gradient shrinking Ricci solitons.

The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.

Regular bundles on orbifolds II: ranks and obstructions

• Authors: Enrique Becerra, Ernesto Lupercio
• arXiv: 2610.02555
• Cross-list: math.AT
• Comments: 29 pages, 2 tables. Part II of arXiv:2609.08125
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT); K-Theory and Homology (math.KT)

A regular complex bundle on an orbispace has fibres that are positive multiples of the stabilizers' regular representations. For stabilizers of order at most two, the sharp uniform rank bound in dimension five is $16$ for paracompact orbispaces and $4$ for closed oriented smooth orbifolds. Rank $16$ occurs with dense free locus and minimum regular rank $4$ on the nonfree locus. We also prove the sharp bound $16$ for closed oriented smooth six-orbifolds. We compute the stable regular index of cyclic gerbes over finite connected CW complexes as the least common multiple of the indices of their character twists. For paracompact orbispaces, optimal integral Adams multipliers give explicit rank bounds in terms of dimension and stabilizer orders. The least positive rank of an integral exterior-power expression with regular fibre and zero reduced class for every regular rank-four bundle on is $32$.

Endomorphisms of Total Kneser Graphs

• Authors: Esmaeil Alhaj Mostafa Abdan, Jacob Russell, Sophia Ulvestad
• arXiv: 2610.03350
• Cross-list: math.CO
• Comments: 13 pages
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

Inspired by work of Gaster, Greene, and Vlamis on the chromatic number of curve graphs, we prove that every endomorphism of the total Kneser graph is an automorphism. We also show that the automorphism group of the total Kneser graph is the product of a pair of permutation groups.

On the type of a complex hyperbolic triangle group

• Authors: Wei Liao, John R Parker
• arXiv: 2610.03422
• Cross-list: math.GR
• Comments: 33 pages
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

In this paper we consider groups of complex hyperbolic isometries generated by three complex involutions. Our first main result states that if the group is discrete and at least two of the mirrors of complex reflections intersect, then the group may be put into a normal form involving the ordering of traces of short, even length words. Our second main result refines a conjecture of Schwartz (proved by Grossi) that determines the type of the group in terms of the angles between the mirrors. That is, in the parameter space of complex triangles with given angles, it determines which of two given words becomes elliptic first.