Explore the fundamental concepts of geometric topology in this 42-minute lecture by Oscar Randal-Williams at the Hausdorff Center for Mathematics. Delve into the topological group of homeomorphisms of d-dimensional Euclidean space, a crucial element in understanding the distinctions between diffeomorphisms and homeomorphisms of d-dimensional manifolds. Discover recent advancements in comprehending the algebraic topology of this group, with insights into the methods employed and emerging conjectural perspectives. Gain valuable knowledge about the intricate relationship between homeomorphisms and manifold structures, excluding the unique case of 4-dimensional space. Engage with cutting-edge research and theoretical developments in this essential area of mathematical study.