math.GT daily digest: 8 new submissions for 18 August 2026

New submissions

Real Heegaard Floer homology with integral coefficients

• Authors: Ciprian Mircea Bonciocat
• arXiv: 2608.15027
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 86 pages, 15 figures

We treat the question of upgrading Guth-Manolescu's real Heegaard Floer theory to a $\mathbb{Z}/2$-graded invariant defined over $\mathbb{Z}$, specifically for the hat version with one basepoint on each component of the branching link. We provide if-and-only-if obstructions to the existence of relative $\mathbb{Z}/2$-gradings and $\mathbb{Z}$-lifts, in terms of purely homological information associated to the 3-manifold with involution. When the corresponding obstruction vanishes, we study the set of such $\mathbb{Z}$-lifts, which a priori is only an invariant up to a torsor action. The obstruction theory could be of independent interest in various Floer theories more broadly, particularly Lagrangian Floer theory.

Linking invariants of spatial graphs

• Authors: E. Alkin, Yu. Khromin, A. Skopenkov
• arXiv: 2608.15860
• Subjects: Geometric Topology (math.GT); Computational Geometry (cs.CG); Combinatorics (math.CO); History and Overview (math.HO)
• Comments: 17 pages (English) + 14 pages (Russian); many figures. The paper belongs to math.GT because the most closely related papers are arXiv:2006.07342 [math.GT], arXiv:2001.01472 [math.GT], arXiv:2410.09860 [math.GT], arXiv:1805.10237 [math.GT], arXiv:2205.01013 [math.GT]

We recall definitions of linking numbers and Wu--Simon numbers for spatial graphs. We expose a `converse' to the Conway--Gordon--Sachs theorem (i.e. description of linking functions for embeddings ), and some results on Wu--Simon numbers. We conjecture and discuss a generalization of the Conway--Gordon--Sachs theorem to multiple linking. The exposition is based on plane diagrams, so no knowledge of spatial geometry is required.

Turaev-Viro invariants for cusped $3$-manifolds

• Authors: Tianyue Liu, Shuang Ming, Xin Sun, Baojun Wu, Tian Yang
• arXiv: 2608.16560
• Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
• Comments: 37 pages, 1 figure

We define a family of Turaev-Viro type invariants for hyperbolic $3$-manifolds with cusp ends, extending the invariants introduced in \cite{LMSWY} for hyperbolic $3$-manifolds with totally geodesic boundary. These invariants are constructed from what we call the ideal -$6j$ symbols, which are variants of the $6j$-symbols associated with the positive representations of the modular double of . We also prove that these invariants decay exponentially, with the exponential decay rate determined by the hyperbolic volume of the manifold.

Cross submissions

Free boundary minimal annuli in convex balls

• Cross-list from: math.DG
• Authors: Dongyeong Ko, Guanhua Shao
• arXiv: 2608.14679
• Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Geometric Topology (math.GT)
• Comments: 72 pages, 1 figure, comments are welcome!

We construct a $4$-parameter family of properly embedded genus-zero surfaces with at most two boundary components in a Riemannian $3$-ball. The construction is a free boundary analog of the canonical $5$-parameter family of surfaces on the $3$-sphere by Marques-Neves in the proof of Willmore conjecture. In the Euclidean unit ball, this family realizes the critical catenoid as a Simon-Smith min-max limit. As applications, we give an upper bound of Almgren-Pitts $4$-width by the area of the critical catenoid. Moreover, we prove the existence of at least three free boundary minimal annuli in any compact $3$-manifold with nonnegative Ricci curvature and strictly convex boundary.

Critical-point-free energy for fractional-Toledo representations

• Cross-list from: math.DG
• Authors: Rivu Bardhan, Anu Dhochak, Pradip Kumar
• arXiv: 2608.15714
• Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Geometric Topology (math.GT)
• Comments: 14 pages

Let be a closed oriented surface of genus . For a reductive representation \rho:\pi_1(S_g)\to\PU(2,1)!, let be the energy function on Teichmüller space associated to equivariant harmonic maps into \CH^2!. For every positive integer $d$ with , all sufficiently large $h$, and every , we construct an irreducible reductive representation [ \rho_{g,h,d}:\pi_1(S_g)\to\PU(2,1) ] with [ \tau(\rho_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{\rho_{g,h,d}})=\varnothing. ] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.

Extremal mappings of tori, Teichmüller potentials and symmetric-space distance

• Cross-list from: math.DG
• Authors: Benson Farb, Eduard Looijenga
• arXiv: 2608.15794
• Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Geometric Topology (math.GT)
• Comments: 15 pages

The symmetric space X_n={\rm SL}(n,\Rb)/{\rm SO}(n)! can be interpreted as the Teichmüller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) {\rm SL}(n,\Rb)!-invariant metric . In 1939 Teichmüller gave a modular interpretation of (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for for has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work.

In this paper we give such a modular interpretation, two in fact. We introduce the {\em total expansion} \TE(f)\in [0,\infty]! of a Lipschitz map between Riemannian manifolds, a notion related to the notion of ``$k$-dilatation'' developed by Gromov, Guth and others. For volume-preserving Lipschitz maps f:\Tc_0\to\Tc_1! between $n$-dimensional, flat, unit-volume tori, we prove that \TE(f)! is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value . We prove similar results for the \emph{Hilbert-Schmidt expansion} \HE(f)!, which is a simple integral over $M$ and has more of an flavor.

Borel classification of simplicial complexes and non-compact $2$- and $3$-manifolds

• Cross-list from: math.LO
• Authors: Martina Iannella, Vadim Weinstein
• arXiv: 2608.16400
• Subjects: Logic (math.LO); Geometric Topology (math.GT)
• Comments: 59 pages

We generalize the Stone space of ultrafilters on Boolean algebras and prove a generalization of Stone duality which is applicable to locally compact Polish spaces. Using this, we obtain complete invariants for simplicial complexes up to PL-homeomorphism and for non-compact $2$- and $3$-manifolds up to homeomorphism. We prove that the homeomorphism relation on non-compact $2$-manifolds without boundary, the homeomorphism relation on non-compact $3$-manifolds with or without boundary, the homeomorphism relation on open subsets of and , and conjugacy of Cantor sets in are classifiable by countable structures. Together with known lower bounds, this implies that these relations are Borel bireducible with isomorphism of countable graphs. We also show that PL-homeomorphism of Heine-Borel simplicial complexes and PL-homeomorphism of PL $n$-manifolds, for every $n$, are classifiable by countable structures.

Computing the $U$ maps on ECH of prequantization bundles

• Cross-list from: math.SG
• Authors: Guanheng Chen
• arXiv: 2608.16625
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)
• Comments: Comments are welcome!

The purpose of this paper is to study the $U$-module structure of the embedded contact homology of a class of prequantization circle bundles over closed symplectic surfaces. We assume that the Euler number of each circle bundle is strictly less than the Euler characteristic of the corresponding surface. We give an explicit description of the $U$-map with respect to a basis of filtered ECH defined via cobordism maps. We use the formula for the $U$-map to compute the ECH spectrum and, as an application, the ECH capacities of certain complements of symplectic divisors. Furthermore, we show that the ECH capacities of closed integral symplectic manifolds are infinite.