math.GT daily digest: 11 submissions for 21 October 2026

New submissions

Orthogonal group sections via light bulb systems

• Authors: Yi Liu
• arXiv: 2610.27126
• Comments: 34 pages; comments welcome
• Subjects: Geometric Topology (math.GT)

For any oriented, smooth $4$--manifold $X$ diffeomorphic to , this paper shows that the natural group homomorphism $$\mathrm{Mod}(X,\partial X)\to \mathrm{O}(H_2(X;\mathbb{Z}))$$ admits a section. In fact, every light bulb system in $X$ gives rise to a section, which can be characterized uniquely in terms of implanted barbell twists and implanted crossbell involutions. Moreover, all sections arising from different light bulb systems become identical after a single --stabilization of $X$.

Weak reducibility for bridge trisections

• Authors: Samuel Lewis-Monkman
• arXiv: 2610.27382
• Comments: 22 pages, 25 figures. Comments welcome
• Subjects: Geometric Topology (math.GT)

A bridge splitting of a knot is weakly reducible if it has a pair of disjoint shadows, one from above and one from below, in the splitting surface. The process of weak reduction exchanges local minima and maxima in the splitting, untelescoping the splitting and giving a method for detecting essential surfaces in the knot complement. Bridge trisections generalize bridge splittings to the setting of surfaces in 4-manifolds. We define weak reducibility in this context, giving diagrammatic conditions for carrying out a weak reduction operation on bridge splittings. We produce a partial classification of 4-bridge trisections of surfaces in .

U torsion structure in Heegaard Floer homology

• Authors: Yuhui Chen, Shunyu Wan
• arXiv: 2610.27646
• Comments: 35 pages, 2 figures
• Subjects: Geometric Topology (math.GT)

We construct infinite families of Brieskorn spheres with three singular fibers for which the difference between the largest and the second largest length of U-torsion in their Heegaard Floer homology increases to infinity as the family parameter increases. In particular, these families give a negative answer to the strong geography restriction on rational homology spheres. As an application, these families give the examples of integer homology spheres that are neither surgery on a knot nor large surgery on links.

Geometric chains of orbifolds with corners and singular homology

• Authors: Hao Yu
• arXiv: 2610.28446
• Subjects: Geometric Topology (math.GT)

We study a geometric homology theory (GHT for short). Its chains are generated by smooth maps from compact oriented orbifolds with corners to a smooth space, and two kinds of chains are set to zero: those admitting an orientation reversing automorphism, and degenerate ones, which, together with their boundaries, factor through smooth maps to orbifolds of lower dimension. Bisecting the source of a geometric chain along a hypersurface transverse to its corners changes the chain only by the boundary of an explicit prism and by prisms over its faces; cutting a geometric cycle along a regular level set of a smooth function on the target does not change its homology class. Geometric chains also behave well under products and pullbacks along fibre bundles, which overcomes some of the combinatorial rigidity of singular chains. We prove that geometric homology is naturally isomorphic to singular homology with rational coefficients for every smooth space admitting a good cover, such as smooth manifolds, orbifolds and locally finite polyhedra, and also for orbifolds with corners. The proof uses cutting operators along level sets of functions on the target, realized by explicit prisms, instead of triangulations. For continuous maps into arbitrary spaces we conjecture the same statement, and prove it for Euclidean neighbourhood retracts in degrees at most one, with surjectivity in degree two. Without the relation killing degenerate chains the comparison fails for orbifold chains: the weighted projective plane , prime, mapped to a point is a cycle that does not bound. Applications on Morse theory and the BV structure on geometric chains of moduli spaces of bordered Riemann surfaces, following Costello and Harrelson--Voronov--Zúñiga, are given.

Cross submissions

Disc-presheaves and normal invariants

• Authors: Oliver H. Wang
• arXiv: 2610.26895
• Cross-list: math.AT
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

In this note, we elaborate on a result of Krannich--Kupers--Mehzer on Disc-presheaves. We show that equivalences of $2$-truncated Disc-presheaves of manifolds yields homotopy equivalences with trivial normal invariant. As an application, we show that two closed, simply connected manifolds are homeomorphic if and only if their $2$-truncated Disc-presheaves are equivalent.

Flux Vacua, Asymptotic Periods, and Relatively Anosov Representations

• Authors: Paul Balavoine, Severin Lüst
• arXiv: 2610.27077
• Cross-list: hep-th
• Comments: 66 pages, 5 figures
• Subjects: High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We develop a topological method to detect classical flux vacua of M-theory on Calabi-Yau fourfolds with a single complex structure modulus using only asymptotic period data and monodromy information. To this end, we express the index of the F-term of the flux superpotential as a winding number along the boundary of the universal cover of the complex structure moduli space. The index evaluates to a sum over zeros and poles of a suitably normalized boundary F-term, where local contributions can be computed algebraically on a dense set of boundary points. Assuming a conjectured bound on the remaining contributions, sufficiently many sign changes of the boundary F-term guarantee a flux vacuum in the interior. We illustrate the method on the mirror Fermat sextic with explicit examples and a systematic flux scan.

A main technical ingredient in our construction is a continuity result for projective period maps: we prove that the non-holomorphic (3,1)-period map extends continuously to the boundary of the covering space if the monodromy representation is relatively Anosov. In general, this property does not hold for holomorphic (4,0)-periods.

Tree Lattices and the Neumann-Zagier conjecture

• Authors: Joshua Evan Greene, Ahmad Issa, Duncan McCoy
• arXiv: 2610.27095
• Cross-list: math.CO
• Comments: 8 pages
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

A tree with integer weights on its vertices presents an integral lattice called a tree lattice. We prove that if a tree lattice is positive definite, then one of the vertex elements is irreducible, i.e. determines a facet of its Voronoi polyhedron. As a corollary, we settle a 1985 conjecture of Neumann and Zagier: if a tree lattice is isometric to the lattice of integer points in Euclidean space, then the tree contains a vertex of weight one.

Maximal ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems

• Authors: Jan Kim, Junseok Kim
• arXiv: 2610.27716
• Cross-list: math.GR
• Comments: 50 pages, comments are welcome!
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We determine the maximum ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems. To obtain the upper bounds, we use the cubical flat torus theorem of Wise and Woodhouse. We then verify the sharpness of these upper bounds by constructing free abelian subgroups of the corresponding ranks. Our result shows that the maximal ranks of free abelian subgroups of ordinary cactus groups can be larger than conjectured by Genevois. As an application, we show that these groups are not quasi-isometric to any right-angled Artin group unless they are virtually free abelian.

Classification of finite subgroups of generalized cactus groups

• Authors: Junseok Kim
• arXiv: 2610.27721
• Cross-list: math.GR
• Comments: 44 pages, comments are welcome!
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We classify the finite groups that embed into generalized cactus groups associated with arbitrary Coxeter systems $(W,S)$. Every finite subgroup is conjugate to a subgroup of for some spherical subset . This reduces the classification to explicit lists of finite groups for the finite irreducible Coxeter systems. For types $B$, $D$, $E$, $F$, and $H$, we analyze cube stabilizers to obtain these lists. For type , we determine directly from its defining presentation. We explicitly realize every listed group as a subgroup of the corresponding generalized cactus group. When $S$ is finite, we determine the largest order and all possible orders of finite subgroups of . In type $A$, we prove directly that a finite group embeds into the ordinary cactus group if and only if it embeds into a Sylow $2$-subgroup of the symmetric group . We also give weighted embedding criteria in types $A$ and $B$ and counterexamples to Genevois's conjecture on finite subgroups of cactus groups.

Median spaces with the strict separation property and superrigidity of actions

• Authors: Indira Chatterji, Lamine Messaci
• arXiv: 2610.27866
• Cross-list: math.MG
• Subjects: Metric Geometry (math.MG); Group Theory (math.GR); Geometric Topology (math.GT)

We introduce strictly separated median spaces, prove structural results and establish superrigidity for actions of higher rank lattices on the latter. We consider two different approach, the first one relies on the construction of a boundary map and the second one is self-contained and relies on the median geometry of the space of sections taking values in a median space.

Towards differential expansion for generic representations

• Authors: Ya.Kononov, A.Morozov
• arXiv: 2610.28531
• Cross-list: hep-th
• Comments: 13 pages
• Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT)

We report the first results of phenomenological analysis, based on extended set of explicitly calculated colored HOMFLY-PT polynomials in \cite{knotebook}. They concern differential expansion (DE), which separates the dependencies on the knot $\mathcal{K}$ and the representation $R$. %of colored HOMFLY-PT polynomials. DE is well understood for symmetric and rectangular representations, while for non-rectangular $R$ even the knot-independent basis of DE is not known. We concentrate on two issues: the possible derivation of DE and on improved characteristic of its defect. We discuss how evolution families -- twist knots, double braids and 3-pretzels -- can be used to find DE by a peculiar triangular transformation of evolution coefficients. Once are found for particular family, they are expected to stay basically the same for arbitrary knots -- modulo controllable reductions, characterized by the defect of DE, which puts knots into different universality classes. Contrary to original expectation, these classes appear labeled not by a single number , but by a whole non-decreasing sequence , which is squeezed between two staircases, , where the upper bound involves Alexander polynomial and follows from the continuation of the Melvin--Morton--Rozansky-Kashaev limit to with , when the limit gives a {\it positive} power of Alexander. The old assumption is oversimplified and fails for 19 non-alternating knots with 11 and 12 crossings. Last, we list the differences of non-rectangularly colored HOMFLY for various mutant pairs to demonstrate their universality (knot-independence) and nice consistency with the spirit of DE.