math.GT daily digest: 6 new submissions for 13 July 2026

Locally Euclidean Discretized Graph Configuration Spaces

• Authors: Justin Murray; Zach Wilcher
• arXiv: 2607.08810 2
• Subjects: Geometric Topology (math.GT)
• Comments: 29 pages, 35 figures. This work is based on the second author's Master's Thesis at Ball State University

We investigate discretized graph configuration spaces as defined by Aaron Abrams in his PhD thesis in 2000. Specifically, we extend Abrams' work by completely classifying which ones admitted by simple graphs are homeomorphic to closed manifolds. In the process of doing so, we develop techniques to translate topological properties of these spaces into graph theoretic ones.

Singularity of Cannon-Thurston maps

• Authors: Vaibhav Gadre; Joseph Maher; Catherine Pfaff; Caglar Uyanik
• arXiv: 2607.08923 3
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: to appear in Trans. Amer. Math. Soc., split off from arXiv:2510.04350, 54pp

In a closed fibered hyperbolic 3-manifold M ,the inclusion of a fiber S, with S and M lifted to the universal covers, gives an exponentially distorted embedding of the hyperbolic plane into hyperbolic 3-space. Nevertheless, Cannon and Thurston showed that there is a map from the circle at infinity of the hyperbolic plane to the 2-sphere at infinity of hyperbolic 3-space. The Cannon-Thurston map is surjective, finite-to-one, and gives a space-filling curve. Here we use properties of geodesics to prove that many natural measures on the circle when pushed forward by the Cannon-Thurston map become singular with respect to many natural measures on the 2-sphere. The circle measures we consider are the Lebesgue measure and stationary measures that arise from fully supported random walks on the surface group. The measures on the sphere we consider are the Lebesgue measure and stationary measures that arise from geometric random walks on the 3- manifold group. We obtain the singularity of measures from the following properties of typical geodesics. We prove that a hyperbolic geodesic sampled with respect to a pushforward measure asymptotically spends a definite proportion of its time close to a fiber. On the other hand, we show that a hyperbolic geodesic sampled with respect to a natural measure on the sphere spends an asymptotically negligible proportion of its time close to a fiber. For a more restricted class of circle measures, namely the Lebesgue measure and stationary measures from geometric random walks on the surface group, we also prove an effective result for the proportion of time spent close to a fiber.

Topological line arrangements with high multiplicities

• Authors: Paolo Aceto; Marco Golla
• arXiv: 2607.09398 4
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Combinatorics (math.CO)
• Comments: 13 pages. Comments are welcome

We investigate constraints on the existence of topological and smooth realisations of combinatorial line arrangements and -configurations in the complex projective plane. By replacing complex lines with locally-flatly or smoothly embedded 2-spheres, we explore the extent to which classical geometric results, such as Hirzebruch's inequality, persist in the topological or smooth category. We introduce two classes of special line arrangements that we call odd and even. We provide constraints for any smoothly realised, non-trivial, odd arrangement via Furuta's 10/8-Theorem. By looking at branched double covers and using the G-signature theorem, we study topologically realised, non-trivial, even arrangement. Finally, we establish a new lower bound for -configurations, showing that for any topologically realised configuration we have , which implies the non-existence of topological realisations for finite projective planes.

Smooth Realizations of Line Configurations

• Authors: Paolo Aceto; Duncan McCoy; JungHwan Park
• arXiv: 2607.09439 5
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Combinatorics (math.CO)
• Comments: 18 pages. Comments are welcome

We study the problem of realizing line configurations as collections of 2-spheres smoothly embedded in the complex projective plane. Building upon prior work by Ruberman and Starkston on topological realizations, we establish a stronger obstruction in the smooth category. Our proof relies on lattice-theoretic arguments based on Donaldson's diagonalization theorem.

Homological Topological Quantum Field Theories

• Authors: Aleksei Andreev
• arXiv: 2607.09601 6
• Subjects: Geometric Topology (math.GT)
• Comments: 53 pages

We develop a new framework for quantum invariants of $3$-manifolds by extending to cobordisms a homological construction of mapping class group representations. More specifically, we construct a -dimensional topological quantum field theory (TQFT) that assigns to each surface the twisted homology of its unordered configuration space. The construction requires a choice of local systems on configuration spaces together with additional data. We formulate sufficient conditions on these data that guarantee the TQFT axioms, and we show that there are at least two useful examples satisfying them. One of them yields a homological construction of the projective Kerler--Lyubashenko TQFT, while the other recovers the Frohman--Nicas--Donaldson TQFT. In contrast to the classical algebraic constructions of quantum invariants, our approach is purely topological and relies on multi-trajectory spaces of cobordisms.

On the density and surjectivity of $\mathbf{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations

• Authors: Renaud Detcherry; Pierre Godfard; Ramanujan Santharoubane
• arXiv: 2607.09633 7
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 79 pages, 4 figures

In this paper, we establish several new fundamental properties of $\mathrm{SO}(3)$-quantum representations of mapping class groups of surfaces, at prime-order roots of unity. We show that for any surface of genus , any number of punctures, and any coloration of the punctures, has dense image in the projective unitary group , extending a landmark result of Larsen and Wang. Moreover, we show that the representations are surjective modulo any unramified maximal ideal of , establishing an effective version of strong approximation for these representations. We also give several applications of our main results to residual finite simpleness of (answering a question of Masbaum and Reid); to subnormal cores of some subgroups of ; to realizability of congruence classes of quantum invariants; to embedding obstructions between $3$-manifolds; and to homological stability for mapping class groups with coefficients in $\mathrm{SO}(3)$-quantum representations.