my-server
← Wiki

Local system

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 is a local system if every point has an open neighborhood 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 of abelian groups has the same stalk 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 is called the monodromy representation of .

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 (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 to the category of modules over a commutative ring , where typically . This is equivalently the data of an assignment to every point a module along with a group representation such that the various are compatible with change of basepoint and the induced map on fundamental groups.

Examples

  • Constant sheaves such as . This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:
  • Let . Since , there is an family of local systems on X corresponding to the maps :
  • Horizontal sections of vector bundles with a flat connection. If is a vector bundle with flat connection , then there is a local system given by For instance, take and , the trivial bundle. Sections of E are n-tuples of functions on X, so defines a flat connection on E, as does for any matrix of one-forms on X. The horizontal sections are then</p> i.e., the solutions to the linear differential equation .<p>If extends to a one-form on the above will also define a local system on , so will be trivial since . So to give an interesting example, choose one with a pole at 0:</p> in which case for ,
  • An n-sheeted covering map is a local system with fibers given by the set . Similarly, a fibre bundle with discrete fibre is a local system, because each path lifts uniquely to a given lift of its basepoint. (The definition adjusts to include set-valued local systems in the obvious way).
  • A local system of k-vector spaces on X is equivalent to a k-linear representation of .
  • If X is a variety, local systems are the same thing as D-modules which are additionally coherent O_X-modules (see O modules).
  • If the connection is not flat (i.e. its curvature is nonzero), then parallel transport of a fibre F_x over x around a contractible loop based at x_0 may give a nontrivial automorphism of F_x, so locally constant sheaves can not necessarily be defined for non-flat connections.

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.

  • Given a locally constant sheaf of abelian groups on X, we have the sheaf cohomology groups with coefficients in .
  • Given a locally constant sheaf of abelian groups on X, let be the group of all functions f which map each singular n-simplex to a global section of the inverse-image sheaf . These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define to be the cohomology of this complex.
  • The group of singular n-chains on the universal cover of X has an action of by deck transformations. Explicitly, a deck transformation takes a singular n-simplex to . Then, given an abelian group L equipped with an action of , one can form a cochain complex from the groups of -equivariant homomorphisms as above. Define to be the cohomology of this complex.

If X is paracompact and locally contractible, then . If 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 is a sheaf 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 . For example, if we look at the complex points of the morphism

then the fibers over

are the plane curve given by , but the fibers over are . If we take the derived pushforward then we get a constructible sheaf. Over we have the local systems

while over we have the local systems

where 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.

See also

References

External links

  • Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.