Local system

In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group "A", and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space "X". Such a concept was introduced by Norman Steenrod in 1943.
Definition.
Let "X" be a topological space. A local system (of abelian groups/modules/...) on "X" is a locally constant sheaf (of abelian groups/modules...) on "X". In other words, a sheaf formula_1 is a local system if every point has an open neighborhood formula_2 such that formula_3 is a constant sheaf.
Path-connected spaces.
If "X" is path-connected, a local system formula_1 of abelian groups has the same fibre "L" at every point. To give such a local system is the same as to give a homomorphism
and similarly for local systems of modules... The map formula_6 giving the local system formula_1 is called the monodromy representation of formula_1.
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.
Stronger definition on non-connected spaces.
Another (stronger, nonequivalent) definition generalising 2, and working for non-connected "X", is: a covariant functor
from the fundamental groupoid of formula_10 to the category of modules over a commutative ring formula_11. Typically formula_12. What this is saying is that at every point formula_13 we should assign a module formula_14 with a representation formula_15 such that these representations are compatible with change of basepoint formula_16 for the fundamental group.
Examples.
If the connection is not flat, parallel transporting a fibre around a contractible loop at "x" may give a nontrivial automorphism of the fibre at the base point "x", so there is no chance to define a locally constant sheaf this way.
The Gauss–Manin connection is a very interesting example of a connection, whose horizontal sections occur in the study of variation of Hodge structures.
Generalization.
Local systems have a mild generalization to constructible sheaves. A constructible sheaf on a locally path connected topological space formula_10 is a sheaf formula_1 such that there exists a stratification of
where formula_49 is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map formula_50. For example, if we look at the complex points of the morphism
then the fibers over
are the smooth plane curve given by formula_53, but the fibers over formula_54 are formula_55. If we take the derived pushforward formula_56 then we get a constructible sheaf. Over formula_54 we have the local systems
while over formula_52 we have the local systems
where formula_61 is the genus of the plane curve (which is formula_62).
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.