  We construct the Fukaya category of a surface with genus greater than one and
compute its Grothendieck group. We consider here a topological variant, in
which we disregard the area form and use instead an admissibility condition
borrowed from Heegaard-Floer theory which ensures invariance under isotopy. We
also study action of the mapping class group of the surface on its Fukaya
category and establish faithfulness.
