代数拓扑：2025–2026 年度综述

An annual survey of the math.AT arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

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

• Equivariant and Motivic Stable Homotopy Theory (15.5%) — This subfield focuses on equivariant cohomology, motivic spectra, and tensor-triangular geometry, exploring algebraic structures within stable homotopy categories.
• Higher Categories and Operadic Algebra (12.5%) — Research centers on infinity-categories, operads, and derived algebraic structures, aiming to unify Koszul dualities and model higher categorical limits.
• Topological Field Theory and Gauge Geometry (11.1%) — This area investigates topological signatures in quantum field theories, including Chern-Simons theories, higher gauge fields, and their connections to cohomotopy.
• Braid Groups and Fundamental Group Conjectures (10.0%) — Studies focus on the algebraic and topological properties of braid groups and fundamental groups, addressing conjectures related to Artin groups and homotopy groups.
• Persistent Homology and Computational Topology (9.1%) — This subfield develops algorithms and theoretical frameworks for persistent homology, groupoid homology, and topological data analysis in computational settings.
• Geometric Topology of Manifolds and Tori (8.8%) — Research examines the geometry of manifolds, moment-angle complexes, and toric varieties, focusing on homotopy types and geometric purity in high-dimensional spaces.
• Multiparameter Persistence and Stability Theorems (8.8%) — This area advances the theory of multiparameter persistence modules, establishing stability results and computational methods for persistence diagrams.
• Configuration Spaces and Moduli Spaces (8.1%) — Investigates the topology of configuration spaces and moduli spaces, including framed configurations, exotic spheres, and infinite braid groups.
• Stable and Rational Homotopy Theory (8.0%) — Focuses on stable homotopy types, rational homotopy theory, and spectral moduli problems, utilizing point-set models and obstruction sequences.
• Simplicial Complexes and Discrete Topology (8.0%) — Explores the combinatorial and geometric properties of simplicial complexes, Vietoris-Rips complexes, and discrete homotopy hypotheses in graph theory.

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.