I overlooked some early posts to this Subject, so I'm not sure if this was mentioned yet: 7 is the lowest dimension d *known* for which the sphere S^d of d dimensions (all points at a distance of 1 from the origin in R^(d+1) has more than one inequivalent differentiable structure. (This was discovered by John Milnor in 1956, in his coyly titled paper "On manifolds homeomorphic to the 7-sphere". At that time it had been expected that each topological manifold admits a unique differentiable structure. Later work showed that S^7 has exactly 28 inequivalent (classes of) differentiable structures. A rumored connection with Howard Johnson ice cream flavors has never been proven.) The only possible exception is the 4-sphere, but no one knows at this time whether it supports a differentiable structure other than the standard one. --Dan