The triangulation conjecture (asking whether a manifold is necessarily a simplicial complex) has been recently resolved in the negative by Ciprian Manolescu. His proof is based on work of Galweski–Stern and Matumoto, reducing the problem to three- and four-dimensional topology. Manolescu solved the low- dimensional problem by developing a new version of Floer homology, resting on the Seiberg–Witten equa- tions and a symmetry of these equations. The resulting Pin(2)-equivariant theory turned out to be a rich source of invariants, and similar ideas have been applied in Heegaard Floer homology. In the lecture we intend to put the problems into context, indicate the solution of Manolescu and draw attention to further developments based on these ideas.