01 · The idea
In everyday language
To simplify a manifold, topologists often use disks to remove intersections between surfaces. In dimension four this is delicate because two-dimensional surfaces meet in isolated points, and attempts to remove intersections can create new ones. Freedman developed methods that control infinite constructions and produce the topological disks needed for the argument. For closed simply connected four-manifolds, intersection forms, together with a further invariant, then organise the homeomorphism classification. One consequence identifies a homotopy four-sphere with the usual sphere topologically. The classification concerns continuous topology: its coordinate changes need not be differentiable, and it leaves separate smooth questions untouched.
Why it matters
Freedman’s methods solved the topological four-dimensional Poincaré problem and transformed manifold classification, while comparison with smooth results revealed the exceptional character of dimension four.
