Triangulations of Manifolds and the Triangulation Conjecture

Explore the fascinating world of topological manifolds and the triangulation conjecture in this 49-minute lecture by Ciprian Manolescu. Delve into the history of the conjecture, which posits that any n-dimensional topological manifold is homeomorphic to a simplicial complex. Learn why this conjecture holds true for dimensions up to 3 but fails in dimension 4, as demonstrated by the work of Casson and Freedman. Discover the proof that disproves the conjecture in higher dimensions, building upon the foundational work of Galewski-Stern and Matumoto. Understand how they reduced the problem to a question in low dimensions, specifically the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group. Explore the use of Pin(2)-equivariant Seiberg-Witten Floer homology, a variant of Floer homology, to answer this low-dimensional question in the negative, ultimately disproving the triangulation conjecture for higher dimensions.