The triangulation conjecture stated that any n-dimensional topological manifold has a homeomorphism to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this series of talks I will explain the proof that it is also false in higher dimensions. This is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. From Floer homology one can extract an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. This is an analogue of Froyshov's correction term in the Pin(2)-equivariant setting.