Local System - Applications

Applications

Examples arise geometrically from vector bundles with flat connections, and from topology by means of linear representations of the fundamental group.

The cohomology with local coefficients in the module corresponding to the orientation covering can be used to formulate Poincaré duality for non-orientable manifolds: see Twisted Poincaré duality.

Larger classes of sheaves are useful: for example the idea of a constructible sheaf in algebraic geometry. These turn out, approximately, to be local coefficients away from a singular set.