Homeomorphisms vs. Diffeomorphisms * Liviu I. Nicolaescu

Consider the situation in Figure 1. The radial projection maps the boundary of the square homeomorphically onto the circle. The net effect of this transformation was to smooth out the corners, while leaving the topology unchanged. We can then ask if we can smooth out the corners of any topological manifold, and if so, are there many different ways of doing this. This turns out to be a very subtle problem, and even some of the simplest situations lead to completely unexpected conclusions. I would like to give you a glimpse of this developing line of research. I will begin by rigourously defining what it means to “smooth out the corners”, and explain when we should consider two smoothing procedures to be inequivalent. Next I will describe a sample of known facts.