Mapping class group, codes and generalized Kummer surfaces

 In this talk codes are binary codes. They show up in topology as follows. Let M be a closed
3-manifold with involution and finite fixed point set F. Then the image in equivariant chomology
of H^1_{Z/2}(M;Z/2) \to H^1_{Z/2}(F;Z/2) = Z/2^r is a code. Poincare-Lefschetz duality implies
that this is a self dual code. These are codes with very interesting relations to positive definite
unimodular bilinear forms over Z (and so to modular forms). This relation indicates that it is
not easy to find explicit 3-manifolds with involution with finite fixed set. I will describe a map
from the mapping class group to such 3-manifolds. This gives a big class of such manifolds
and so of self dual binary codes. Amongst the self dual codes there is an interesting subclass,
the doubly even codes. There is a construction of a closed smooth 4-manifold from a
3-manifold with involution which in the special case of the 3-manifold coming from the
identity map on the 2-torus gives the Kummer surface. There is a relation between the property
that this manifold has a spin structure and that the code attached to the 3-manifold is doubly
even. The resulting 4-manifolds from arbitrary elements in the mapping class group are a
sort of generalization of the Kummer surface. Its relevance for codes will be explained.