math.GT daily digest: 9 submissions for 30 September 2026

New submissions

Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds

• Authors: Alberto Verjovsky
• arXiv: 2609.35858
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

Let be the closed hyperbolic $3$--manifold which is the mapping torus of a pseudo-Anosov homeomorphism $f$ of a closed surface $S$ of genus at least two. The inclusion of the fiber subgroup has a Cannon--Thurston boundary map , which is a Peano curve. We relate this map to two variational representatives of the fiber homotopy class: the harmonic map of Eells--Sampson and Hartman, and a least-area embedded minimal surface. Equivariant lifts of homotopic maps are a bounded distance apart; hence, for every marked conformal structure on $S$, the lift of the harmonic representative extends continuously to the closed disk with boundary value the Cannon--Thurston map, and a least-area fiber lifts to a properly embedded minimal plane in \Hh^3! with the same boundary map. We give area and curvature bounds for a least-area fiber, show that its largest principal curvature is at least one, and study the energy of the harmonic representatives as a function on Teichmüller space, its critical points, which are the zeros of the Hopf differential, and its invariance under the monodromy.

On intersection forms of spin $4$-manifolds with AR homology sphere boundary

• Authors: Eiichiro Hakamada
• arXiv: 2609.35931
• Comments: 39 pages, 2 figures
• Subjects: Geometric Topology (math.GT)

In this paper, building on the work of Dai--Sasahira--Stoffregen, we derive constraints on the intersection forms of smooth, compact, indefinite, spin $4$-manifolds bounded by AR homology spheres using two methods. First, we use the constraints on the existence of $\mathrm{Pin}(2)$-equivariant maps between representation spheres, established by Hopkins--Lin--Shi--Xu in their proof of the $10/8+4$-inequality. Second, we compute Lin's $\kappa o$-invariant for AR homology spheres. Combined with Lin's version of the relative $10/8$-inequality, this computation yields constraints different from those based on Hopkins--Lin--Shi--Xu. As applications, we obtain new lower bounds on the minimal genus of surfaces bounded by knots whose double branched covers are AR homology spheres in various $4$-manifolds, and provide an obstruction to smooth $4$-manifold decompositions related to the $11/8$-conjecture.

Higher Alexander Polynomials of Satellite Links

• Authors: Soheil Azarpendar; Colin McCulloch
• arXiv: 2609.36040
• Comments: 47 pages, 3 figures. Comments welcome!
• Subjects: Geometric Topology (math.GT)

We present formulas for the higher Alexander polynomials of satellite links.

Quantitative finiteness of monic characters of knots

• Authors: Taehee Kim; Takayuki Morifuji
• arXiv: 2609.36772
• Comments: 11 pages
• Subjects: Geometric Topology (math.GT)

Dunfield, Friedl, and Jackson showed that if the -character variety of a knot has an irreducible curve component that contains the character of an irreducible representation and a character with nonmonic twisted Alexander polynomial, then this component has only finitely many characters with monic twisted Alexander polynomials. In this paper, we give explicit upper bounds on the number of such characters in terms of a presentation of the knot group. In particular, we give upper bounds in terms of the crossing number of the knot.

On topologically transitive subsets of flag manifolds

• Authors: Subhadip Dey; Sami Douba; Konstantinos Tsouvalas
• arXiv: 2609.37752
• Comments: 16 pages. Comments welcome
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

We discuss some contexts in which the topological dynamics of certain Anosov subgroups of higher-rank Lie groups on "bad" subsets of flag manifolds can be analyzed using results from homogeneous dynamics in the infinite-covolume rank-one setting. For example, we show that a group of projective transformations dividing a strictly convex domain in projective space and intersecting Zariski-densely the stabilizer of an ellipsoid has a dense orbit in the complement of the domain, and acts minimally on the space of full projective flags tangent to the domain. This accounts for all known examples of divisible strictly convex domains in sufficiently high dimensions. We also provide examples of Zariski-dense groups that are Anosov in a partial flag manifold but such that the equivariant projection from the Benoist-Guivarc'h limit set in the Furstenberg boundary to the Anosov limit set is not a fibration.

Exact triangle and spectral sequence of the equivariant Floer homology of double branched covers

• Authors: Jinzhou Huang
• arXiv: 2609.38117
• Comments: 25 pages, 6 figures
• Subjects: Geometric Topology (math.GT)

In [OS05] the authors proved an exact triangle for the Heegaard Floer homology of the double branched covers associated to a skein sequence of links, and constructed a spectral sequence from the reduced Khovanov homology to the Heegaard Floer homology of the double branched cover of a link. For a link and $\Sigma (L)$ its double branched cover, the invariant doesn't capture the information of the natural involution on $\Sigma (L)$. In this paper we study the equivariant version of and prove that similar exact triangle and spectral sequence hold for this equivariant version.

Cross submissions

Farey Symbols for the Picard Group

• Authors: Devendra Tiwari; Helena Verrill
• arXiv: 2609.36629
• Cross-list: math.NT
• Comments: 36 pages, 17 figures
• Subjects: Number Theory (math.NT); Group Theory (math.GR); Geometric Topology (math.GT)

We introduce Picard Farey symbols for finite-index subgroups of G=\PSL_2(\ZZ[i])!, using the Gaussian Farey tessellation of \HH^3! by ideal octahedra. A symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and ordered cooriented face pairings. We prove that a valid marked symbol reconstructs the finite Picard-cell action and hence the subgroup, and we work out a range of exact examples. For a torsion-free subgroup of index $n$, every fundamental domain formed from complete Farey octahedra contains $n/12$ octahedra; a spanning-tree construction gives one with at most side-pairing transformations. We also identify the Gaussian octahedral edge--face complex with the integral rank-two Steinberg presentation for \QQ(i)!. Thus the same decorated geometry directly determines a finite presentation of coefficient-valued Bianchi modular symbols; for we give the complete group-ring boundary matrix. Canonical reduction and an effective Hecke-compatible reduction remain open.

Spin and scalar curvature on punctured four manifolds

• Authors: Aditya Kumar; Boyu Zhang
• arXiv: 2609.37948
• Cross-list: math.DG
• Comments: 13 pages; comments welcome
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We show that a closed spin four-manifold with nonzero signature, even after being punctured, does not admit a complete metric with uniformly positive scalar curvature. In particular, the punctured $K3$ surface does not admit such a metric. This is known to be false in higher dimensions. A key new observation, which could be of independent interest, is that Gromov's $\mu$-bubbles are a convenient class of boundaries for the APS boundary value problem for the Dirac operator.

Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres

• Authors: Fernando Galaz-García
• arXiv: 2609.38126
• Cross-list: math.DG
• Comments: 24 pages
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Sperança's special - bundle models with the compatible-disk construction of He, Liu and Yau. We show that the relevant bundles admit equivariant polar normal forms, with transition functions that are constant along meridians and conjugation-equivariant. We also show that the representation-dependent part of the He--Liu--Yau construction requires only a uniform bound on the infinitesimal action fields. Sperança's $8$- and $10$-dimensional examples satisfy this bound. The resulting northern and southern metrics have matching boundary metrics and compatible second fundamental forms, so the Reiser--Wraith gluing theorem gives the required positively curved metrics.