The Hopf fibration is a map from a sphere in this many dimensions to a sphere in one fewer dimensions.
This is the only dimension for which the smooth Poincare conjecture has not been resolved and the only dimension for which R to the n has an exotic smooth structure. In this many dimensions, isomorphisms of piecewise linear and smooth manifolds are the same, but different from isomorphisms of topological manifolds. The Hopf fibration is a map from a sphere in this many dimensions to a sphere in one fewer dimensions. This is the dimension of an algebra denoted (*) H for its discoverer, Hamilton. The volume of a unit sphere in this many dimensions is pi squared over two. This is the minimum number of dimensions in which the Klein bottle can be embedded with no self-intersections. Minkowski space has this many dimensions since it models spacetime. For 10 points, give the number of dimensions in a tesseract.
This is the only dimension for which the smooth Poincare conjecture has not been resolved and the only dimension for which R to the n has an exotic smooth structure.
four