Based on Fields medal winning work of Michael Freedman， this book explores the disc embedding theorem for 4-dimensional manifolds。 This theorem underpins virtually all our understanding of topological 4-manifolds。 Most famously， this includes the 4-dimensional Poincar� conjecture in the
topological category。

The Disc Embedding Theorem contains the first thorough and approachable exposition of Freedman's proof of the disc embedding theorem， with many new details。 A self-contained account of decomposition space theory， a beautiful but outmoded branch of topology that produces non-differentiable
homeomorphisms between manifolds， is provided， as well as a stand-alone interlude that explains the disc embedding theorem's key role in all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem。 Additionally， the ramifications of the disc embedding
theorem within the study of topological 4-manifolds， for example Frank Quinn's development of fundamental tools like transversality are broadly described。

The book is written for mathematicians， within the subfield of topology， specifically interested in the study of 4-dimensional spaces， and includes numerous professionally rendered figures。