Cutting and pasting varieties using algebraic K-theory
-- Inna Zakharevich, November 13, 2015

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The Grothendieck ring of varieties is defined to be the free abelian
group generated by varieties, modulo the relation that for a closed
subvariety Y of X, [X] = [Y] + [X \ Y]; the ring structure is defined
via the Cartesian product.  For example, if X and Y are piecewise
isomorphic, in the sense that there exist stratifications on X and Y
with isomorphic strate, then [X] = [Y] in the Grothendieck ring.
There are two important questions about this ring: 1.  What does it
mean when two varieties X and Y have equal classes in the Grothendieck
ring?  Must X and Y be piecewise isomorphic?  2.  Is the class of the
affine line a zero divisor?  Last December Borisov answered both of
these questions with a single example, by constructing an element [X]
- [Y] in the kernel of multiplication by the affine line; in a
beautiful coincidence, it turned out that X x A<sup>1</sup> and Y x
A<sup>1</sup> were not piecewise isomorphic.  In this talk we will
describe an approach using algebraic K-theory to construct a
topological version of the Grothendieck ring of varieties.  The
fundamental group of this space is generated by birational
automorphisms of varieties which extend to piecewise automorphisms,
and allows us to construct a group which surjects onto the kernel of
multiplication by the affine line.  By analyzing this group we will
sketch a proof that Borisov's coincidence was not a coincidence at
all: any element in the kernel of multiplication by the affine line
can be represented as [X]-[Y], where X x A<sup>1</sup> and Y x
A<sup>1</sup> are not piecewise isomorphic.