$\mathrm{K3}$: a New Problem List in Low-Dimensional Topology

Nearly 400 intricate problems and a thousand subproblems in low-dimensional topology spur exploration among graduate students and experts alike. Arranged by dimension—covering knot theory, surfaces, 3? and 4?manifolds, and miscellanea—this collection ignites further inquiry and inspires new research directions.

This book is a compendium of problems in low-dimensional topology, each presented with background and references. It is aimed at graduate students and more experienced researchers alike, highlighting the problem-driven nature of the field. The problems are intended to stimulate research and point to new directions in an area that has been extremely active and has broadened tremendously over the past 50 years. The problem list is the outcome of a collaborative, community-driven effort led by R. Inanc Baykur, R. C. Kirby, and D. Ruberman. It began at a week-long workshop at the American Institute of Mathematics in fall 2023 and grew substantially through the involvement of chapter editors, workshop participants, problem proposers, scribes, and referee-altogether drawing contributions from several hundred researchers, from early career mathematicians to senior figures in the field. As in the influential problem lists compiled by R. Kirby in the 1970s and 1990s, the book is organized roughly by dimension: knot theory, surfaces, $3$-manifolds, $4$-manifolds, and a brief miscellanea chapter. It features close to 400 problems across a wide range of topics, with nearly a thousand subproblems, questions, and conjectures interspersed throughout.