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Local system

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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.

Definition

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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

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Path-connected spaces

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If X is path-connected,[clarification needed] a local system of abelian groups has the same stalk 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 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).[2]

Stronger definition on non-connected spaces

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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

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  • 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

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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 .[3] If is the local system corresponding to L, then there is an identification compatible with the differentials,[4] so .

Generalization

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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

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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

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References

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