There is a large difference between smooth and topological properties in four dimensions. The main reason for this is that a pair of two dim'l  surfaces will generically intersect in a collection of points and such intersections may be simplified by a fractal construction that is continuous but not smooth. In particular, there are topologically equivalent (isotopic) spheres that are not smoothly isotopic. One way to see this is via small parts of space known as corks in 4-manifolds.  The study of corks leads to a question in equivariant homology cobordism. We will describe this arc, and sketch a proof of a new result about equivariant homology cobordism.  This is joint work with Kim, Melvin and Ruberman.

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