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.[1]

The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold.[2]

Definition Edit

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 the restricted sheaf   is isomorphic to the sheafification of some constant presheaf.[clarification needed]

Equivalent definitions Edit

Path-connected spaces Edit

If X is path-connected,[clarification needed] a local system   of abelian groups has the same stalk L at every point. There is a bijective correspondence between local systems on X and group homomorphisms


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 to 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.

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   (equivalently,  -modules).[3]

Stronger definition on non-connected spaces Edit

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  , where typically  . This is equivalently the data of an assignment to every point   a module   along with a group representation   such that the various   are compatible with change of basepoint   and the induced map   on fundamental groups.

Examples Edit

  • Constant sheaves such as  . This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:

  • Let  . Since  , there is an   family of local systems on X corresponding to the maps  :

  • Horizontal sections of vector bundles with a flat connection. If   is a vector bundle with flat connection  , then there is a local system given by
    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 fibers given by 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 equivalent to a k-linear representation of  .
  • 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.

Cohomology Edit

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.

  • Given a locally constant sheaf   of abelian groups on X, we have the sheaf cohomology groups   with coefficients in  .
  • Given a locally constant sheaf   of abelian groups on X, let   be the group of all functions f which map each singular n-simplex   to a global section   of the inverse-image sheaf  . These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define   to be the cohomology of this complex.
  • The group   of singular n-chains on the universal cover of X has an action of   by deck transformations. Explicitly, a deck transformation   takes a singular n-simplex   to  . Then, given an abelian group L equipped with an action of  , one can form a cochain complex from the groups   of  -equivariant homomorphisms as above. Define   to be the cohomology of this complex.

If X is paracompact and locally contractible, then  .[4] If   is the local system corresponding to L, then there is an identification   compatible with the differentials,[5] so  .

Generalization Edit

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 Edit

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 Edit

References Edit

  1. ^ Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics. 44 (4): 610–627. doi:10.2307/1969099. MR 0009114.
  2. ^ MacPherson 1990, Theorem 3.8.
  3. ^ Milne, James S. (2017). Introduction to Shimura Varieties. Proposition 14.7.
  4. ^ Bredon, Glen E. (1997). Sheaf Theory, Second Edition, Graduate Texts in Mathematics, vol. 25, Springer-Verlag. Chapter III, Theorem 1.1.
  5. ^ Hatcher, Allen (2001). Algebraic Topology, Cambridge University Press. Section 3.H.

External links Edit

  • "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 (December 15, 1990). "Intersection homology and perverse sheaves" (PDF).
  • El Zein, Fouad; Snoussi, Jawad. "Local systems and constructible sheaves" (PDF).