1-manifold: homeomorphic to a line or a circle;
2-manifold: Classification Theorem of (Closed) Surfaces--closed surfaces must be either spheres, connected sum of tori, or connected sum of projective planes. The former two are orientable and are classified by genus, while the latter is unorientable.
3-manifold: Moise Theorem--every topological 3-manifold has a unique piecewise-linear structure and smooth structure. Thurston's geometrization theorem.
Physics: Chern-Simons-Jones polynomials; Gukov-Pei-Putrov-Vafa (GPPV) invariant-6d (2,0) theory and its compactifications.
4-manifold: open problem when involving smooth structures.
eg. Donaldson invariant, Seiberg-Witten invariant, intersection theory.
(others and $\ge$ 5d: h-cobordism theorem and surgery theory)

categorification, 3d-3d correspondence