In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. Local systems are the building blocks of more general tools, such as constructible and perverse sheaves.

Definition

Let X be a topological space. A local system (of abelian groups/modules...) on X is a locally constant sheaf (of abelian groups/of modules...) on X. In other words, a sheaf \mathcal{L} is a local system if every point has an open neighborhood U such that the restricted sheaf is isomorphic to the sheafification of some constant presheaf.

Equivalent definitions

Path-connected spaces

If X is path-connected, a local system \mathcal{L} of abelian groups has the same stalk L at every point. There is a bijective correspondence between local systems on X and group homomorphisms and similarly for local systems of modules. The map giving the local system \mathcal{L} is called the monodromy representation of \mathcal{L}. This shows that (for X path-connected) a local system is precisely a sheaf whose pullback to the universal cover of X is a constant sheaf. This correspondence can be upgraded to an equivalence of categories between the category of local systems of abelian groups on X and the category of abelian groups endowed with an action of \pi_1(X,x) (equivalently, -modules).

Stronger definition on non-connected spaces

A stronger nonequivalent definition that works for non-connected X is the following: a local system is a covariant functor from the fundamental groupoid of X to the category of modules over a commutative ring R, where typically. This is equivalently the data of an assignment to every point x\in X a module M along with a group representation such that the various \rho_x are compatible with change of basepoint x \to y and the induced map on fundamental groups.

Examples

Cohomology

There are several ways to define the cohomology of a local system, called cohomology with local coefficients, which become equivalent under mild assumptions on X. If X is paracompact and locally contractible, then. If \mathcal{L} is the local system corresponding to L, then there is an identification compatible with the differentials, so.

Generalization

Local systems have a mild generalization to constructible sheaves -- a constructible sheaf on a locally path connected topological space X is a sheaf \mathcal{L} such that there exists a stratification of where is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map f:X \to Y. For example, if we look at the complex points of the morphism then the fibers over are the plane curve given by h, but the fibers over are. If we take the derived pushforward then we get a constructible sheaf. Over \mathbb{V} we have the local systems while over we have the local systems where g is the genus of the plane curve (which is ).

Applications

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.