Wikipedia / Local system
Wikipedia / 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 is isomorphic to the sheafification of some constant presheaf.
Equivalent definitions
Path-connected spaces
If X is path-connected, at every point. There is a bijective correspondence between local systems on X and group homomorphisms
is called the monodromy representation.
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-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
Examples
Constant sheaves. This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:
family of local systems on X:
, the trivial bundle. Sections of E are n-tuples of functions on X defines a flat connection on E on X. The horizontal sections are then.. So to give an interesting example, choose one with a pole at 0:
An n. 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.
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.
The Gauss–Manin connection is a prominent example of a connection whose horizontal sections are studied in relation to variation of Hodge structures.
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.
of abelian groups on X, we have the sheaf cohomology.
of abelian groups on X be the group of all functions f which map each singular n-simplex of the inverse-image sheaf to be the cohomology of this complex.
of singular n-chains on the universal cover of X by deck transformations takes a singular n. Then, given an abelian group L to be the cohomology of this complex.
If X is paracompact and locally contractible. is the local system corresponding to L compatible with the differentials,.
Generalization
Local systems have a mild generalization to constructible sheaves such that there exists a stratification of
. For example, if we look at the complex points of the morphism
then the fibers over
we have the local systems
we have the local systems
).
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.