A fundamental idea to understanding topological spaces is decomposing them down into simpler components for easier analysis. The study of decompositions in unstable homotopy theory has a long, rich history and has contributed significantly to the study of manifolds, Poincaré duality complexes, Lie groups, gauge groups, and polyhedral products.This book provides the first comprehensive exploration of how decompositions shape and influence unstable homotopy theory. It presents 40 open problems across a broad range of topics, inviting readers to push the field forward.Requiring only a foundation in ordinary homology, cohomology theory, and basic homotopy theory, this book is accessible to advanced graduate students and serves as an invaluable resource for experts.
