Title: Towards constructing faithful homological representations of surface mapping class groups.

In 2001 S.Bigelow and D.Krammer independently proved that the Lawrence-Krammer representation of the mapping class group of a punctured disk is faithful, and as it is finite dimensional, this proves linearity of the corresponding MCG or, equivalently, of the braid group. This representation is homological, i.e. it is constructed as homology groups of some topological space with a natural action of the MCG and coefficients in some local system. In particular, Lawrence-Krammer representation is the second homology of the two points configuration space of the disc. It is natural to ask whether this approach works in the case of surface mapping class groups. The main difficulty is to find a right local system on the corresponding configuration space. In this talk I will show how any abelian and the simplest non-abelian (Heisenberg) local systems fail to provide a faithful representation. Then I’ll tell about possible candidates and why they are expected to be a good choice for the local system.