Witt groups of sheaves on topological spaces

Abstract

This paper investigates the Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincaré–Verdier duality. We consider two main cases, that of perfect complexes of sheaves on locally compact Hausdorff spaces and that of cohomologically constructible complexes of sheaves on polyhedra. We show that the Witt groups of the latter form a generalised homology theory for polyhedra and continuous maps. Under certain restrictions on the ring R, we identify these constructible Witt groups of a finite simplicial complex with Ranicki’s free symmetric L-groups. Witt spaces are the natural class of spaces for which the rational intersection homology groups have Poincaré duality. When the ring R is the rationals we identify the constructible Witt groups with the 4-periodic colimit of the bordism groups of PL Witt spaces. This allows us to interpret L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology.