# 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.[1]

## 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 is a local system if every point has an open neighborhood such that is a constant sheaf.

### Equivalent definitions

### Path-connected spaces

If *X* is path-connected, a local system 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 giving the local system is called the **monodromy representation** of .

*Proof of equivalence*

Take local system and a loop at *x*. It's easy so show that any local system on is constant. For instance, is constant. This gives an isomorphism , i.e. between *L* and itself.
Conversely, given a homomorphism , consider the *constant* sheaf on the universal cover of *X*. The deck-transform-invariant sections of 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 is the universal covering.

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 to the category of modules over a commutative ring . Typically . What this is saying is that at every point we should assign a module with a representations of such that these representations are compatible with change of basepoint for the fundamental group.

## Examples

- Constant sheaves. For instance, . This is a useful tool for computing cohomology since the sheaf cohomology

- is isomorphic to the singular cohomology of .

- . Since , there are -many linear systems on
*X*, the one given by monodromy representation

- by sending

- Horizontal sections of vector bundles with a flat connection. If is a vector bundle with flat connection , then

- is a local system.
- 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 - i.e., the solutions to the linear differential equation .
- 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*: - in which case for ,

- An
*n*-sheeted covering map is a local system with sections locally 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 the same as a*k*-linear representation of the group . - If
*X*is a variety, local systems are the same thing as*D*-modules that are in addition coherent as*O*-modules.

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 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 smooth 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.

## References

- Steenrod, Norman E. (1943). "Homology with local coefficients".
*Annals of Mathematics*.**44**(4): 610–627. doi:10.2307/1969099. MR 0009114.

## External links

- "What local system really is". Stack Exchange.
- Schnell, Christian. "Computing Cohomology of Local Systems" (PDF). Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.
- Williamson, Geordie. "An illustrated guide to perverse sheaves" (PDF).
- Macpherson, Robert. "Intersection homology and perverse sheaves" (PDF).
- El Zein, Fouad; Snoussi, Jawad. "Local systems and constructible sheaves" (PDF).