几何拓扑：2025–2026 年度综述

An annual survey of the math.GT arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 108 papers and organised into 40 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Quantum Invariants and Braiding in Low-Dimensional Topology (81.9%) — This cluster focuses on constructing quantum invariants, braiding structures, and rigidity properties in low-dimensional topology, with applications to knots, curves, and mapping class groups.
• Homology and Subgroups of Mapping Class Groups (10.0%) — This cluster investigates homology, finite generation, and representation stability of subgroups of mapping class groups and Torelli groups, along with surface subgroups in hyperbolic 3-manifolds.
• Knot and Link Invariants via Diagrams and TQFT (4.9%) — This cluster develops computational and structural invariants of knots and links using diagrammatic methods, TQFT frameworks, and contact geometry, including unknotting and linking invariants.
• Hyperbolic Geometry and Geodesic Lengths on Surfaces (1.7%) — This cluster studies hyperbolic metrics, geodesic lengths, and geometric properties of surfaces and spaces, including diameter bounds and Cannon–Thurston maps.
• Conjectures and Proofs in Geometric Group Theory (0.6%) — This cluster addresses conjectures and proof techniques in geometric group theory, including the K(π,1) conjecture for Artin groups and Baum–Connes type properties.
• Topological Quantum Field Theories and Algebraic Structures (0.4%) — This cluster develops topological field theories, localization properties, and algebraic structures arising from link homology and transformation groups.
• Homotopy Theory and Classifying Spaces of Racks (0.2%) — This cluster explores homotopy equivalences, classifying maps, and fundamental group properties in the context of rack spaces and singular maps.
• Infinite Families of 3-Manifolds and Stein Fillings (0.2%) — This cluster constructs infinite families of contact 3-manifolds, L-space knots, and corks with Stein structures, focusing on existence and classification results.
• Boundary Connectivity and Affine Cactus Groups (0.1%) — This cluster examines connectedness of minimizing cluster boundaries and hyperbolicity properties of affine cactus groups in low-dimensional settings.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.