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

L

{\displaystyle {\mathcal {L}}}

is a local system if every point has an open neighborhood

U

{\displaystyle U}

such that the restricted sheaf

L

|

U

{\displaystyle {\mathcal {L}}|_{U}}

is isomorphic to the sheafification of some constant presheaf.

Locally constant sheaf

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf

F

{\displaystyle {\mathcal {F}}}

on X such that for each x in X, there is an open neighborhood U of x such that the restriction

F

|

U

{\displaystyle {\mathcal {F}}|_{U}}

is a constant sheaf on U. It is also called a local system. When X is a stratified space, a constructible sheaf is roughly a sheaf that is locally constant on each member of the stratification.

A basic example is the orientation sheaf on a manifold since each point of the manifold admits an orientable open neighborhood (while the manifold itself may not be orientable).

For another example, let

X

=

C

{\displaystyle X=\mathbb {C} }

,

O

X

{\displaystyle {\mathcal {O}}_{X}}

be the sheaf of holomorphic functions on X and

P

:

O

X

→

O

X

{\displaystyle P:{\mathcal {O}}_{X}\to {\mathcal {O}}_{X}}

given by

P

=

z

∂

∂

z

−

1

2

{\displaystyle P=z{\partial \over \partial z}-{1 \over 2}}

. Then the kernel of P is a locally constant sheaf on

X

−

{

0

}

{\displaystyle X-\{0\}}

but not constant there (since it has no nonzero global section).