The definition of topological and smooth manifolds. Morse theory, handle decompositions, surgery. Every group can be the fundamental group of a manifold in dimension greater than three. The h-cobordism theorem (without proof), and the Whitney trick. Application: The generalized Poincaré source:

https://disenowebakus.net/en/symbol-character-html conjecture.

Knots and links: Reidemeister moves, Seifert surface and genus, Alexander polynomial, fibred knots, Jones polynomial, prime decomposition, 4-ball genus

3-manifolds: Heegaard decompositions, unique prime decomposition, loop theorem (without proof), lens spaces, Dehn surgery, branched double cover

4-manifolds: Kirby calculus, the intersection form, Freedman’s and Donaldson’s theorems (without proof)