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.