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.
[1] Local systems are the building blocks of more general tools, such as constructible and perverse sheaves.
A local system (of abelian groups/modules...) on X is a locally constant sheaf (of abelian groups/of modules...) on X.
is a local system if every point has an open neighborhood
is isomorphic to the sheafification of some constant presheaf.
[clarification needed] If X is path-connected,[clarification needed] a local system
of abelian groups has the same stalk
There is a bijective correspondence between local systems on X and group homomorphisms and similarly for local systems of modules.
giving the local system
is called the monodromy representation of
It's easy to show that any local system on
gives a local system on X.
Similarly, the deck-transform-ρ-equivariant sections give another local system on X: for a small enough open set U, it is defined as where
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
[2] 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
This is equivalently the data of an assignment to every point
are compatible with change of basepoint
the above will also define a local system on
So to give an interesting example, choose one with a pole at 0: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
is the local system corresponding to L, then there is an identification
Local systems have a mild generalization to constructible sheaves -- a constructible sheaf on a locally path connected topological space
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
If we take the derived pushforward
is the genus of the plane curve (which is
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.