In the 1970s, Galewski-Stern and Matumoto studied the existence and the classification of triangulations on topological manifolds of dimension at least 5. They reduced these problems to questions about the three-dimensional homology cobordism group, Θ3H, and the Rokhlin homomorphism from this group to ℤ/2. The structure of the homology cobordism group is still unknown, but some information can be obtained using tools from gauge theory and symplectic geometry, such as the Seiberg-Witten Floer spectrum and involutive Heegaard Floer homology. I will describe the proof of the existence of non-triangulable high-dimensional manifolds (using gauge theory), and some open problems.
This video is part of Harvard University‘s conference JDG 2017.
