BNPC n-manifolds (n < 5) are Euclidean

In 1981, Gromov asked whether there exist topological manifolds, other than Euclidean space, that admit a metric of non-positive curvature in the sense of Busemann (abbreviated as BNPC spaces). His question has been completely resolved for CAT(0) manifolds. Since CAT(0) spaces are contractible, it follows from the classification of surfaces that every CAT(0) 2-manifold is Euclidean. In dimension 3, combining results of Brown and Rolfsen shows that CAT(0) manifolds are homeomorphic to R3. Recently, Lytchak-Nagano-Stadler proved that CAT(0) 4-manifolds are Euclidean. In dim ≥ 5, Davis-Januszkiewicz constructed examples of non-Euclidean CAT(0) manifolds. In this talk, I will discuss Gromov's question and introduce BNPC spaces. Furthermore, I will explain how to extend the above results to BNPC manifolds, thereby answering his question. This is joint work with Tadashi Fujioka.