Georgia Institute of Technology College of Sciences

The Grothendieck ring of varieties is defined to be the free abelian group generated by k-varieties, modulo the relation that for any closed subvariety Y of a variety X, we impose the relation that [X] = [Y] + [X \ Y]; the ring structure is defined by [X][Y] = [X x Y]. Last December two longstanding questions about the Grothendieck ring of varieties were answered: 1. If two varieties X and Y are piecewise isomorphic then they are equal in the Grothendieck ring; does the converse hold? 2. Is the class of the affine line a zero divisor? Both questions were answered by Borisov, who constructed an element in the kernel of multiplication by the affine line; coincidentally, the proof also constructed two varieties whose classes in the Grothendieck ring are the same but which are not piecewise isomorphic. In this talk we will investigate these questions further by constructing a topological analog of the Grothendieck ring and analyzing its higher homotopy groups. Using this extra structure we will sketch a proof that Borisov's coincidence is not a coincidence at all: that any element in the annihilator of the Lefschetz motive can be represented by a difference of varieties which are equal in the Grothendieck ring but not piecewise isomorphic.