Local system explained

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

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

l{L} is a local system if every point has an open neighborhood U such that the restricted sheaf l{L}| _{U} is isomorphic to the sheafification of some constant presheaf.

Equivalent definitions

Path-connected spaces

If X is path-connected, a local system

l{L} of abelian groups has the same stalk L at every point. There is a bijective correspondence between local systems on X and group homomorphisms\rho: \pi _{1(X,x)} \to Aut (L) and similarly for local systems of modules. The map \pi _{1(X,x)} \to End (L) giving the local system l{L} is called the monodromy representation of l{L} .

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

\pi _{1(X,x)} (equivalently, Z [\pi _{1(X,x)]} -modules).^[2]

Stronger definition on non-connected spaces

A stronger nonequivalent definition that works for non-connected X is: the following: a local system is a covariant functor

l{L}\colon \Pi _{1(X)} \to bf{Mod}(R) from the fundamental groupoid of X to the category of modules over a commutative ring R , where typically R = \Q,\R,\Complex . This is equivalently the data of an assignment to every point x\in X a module M along with a group representation \rho _{x:} \pi _{1(X,x)} \to Aut _{R(M)} such that the various \rho _{x} are compatible with change of basepoint x \to y and the induced map \pi _{1(X,} x) \to \pi _{1(X,} y) on fundamental groups.

Examples

• Constant sheaves such as

\underline{\Q} _{X} . This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:

$$H^k(X,\backslash underline\_X)\; \backslash cong\; H^k\_\backslash text(X,\backslash Q)$$

• Let

X = \R ^{2} \setminus \{( 0 , 0 )\} . Since \pi ^{2} _{1 (\R} \setminus \{( 0 , 0 )\}) = Z , there is an S ^{1} family of local systems on X corresponding to the maps n \mapsto e ^{in\theta} :

$$\backslash rho\_\backslash theta:\; \backslash pi\_1(X;\; x\_0)\; \backslash cong\; \backslash Z\; \backslash to\; \backslash text\_\backslash Complex(\backslash Complex)$$

• Horizontal sections of vector bundles with a flat connection. If

E\to X is a vector bundle with flat connection \nabla , then there is a local system given by $$E^\backslash nabla\_U=\backslash left\backslash$$ For instance, take X = \Complex \setminus 0 and E = X x \Complex ^{n} , the trivial bundle. Sections of E are n-tuples of functions on X, so \nabla _{0(f} _{1,...,f} _{n)=} (df _{1,...,df} _{n)} defines a flat connection on E, as does \nabla(f _{1,...,f} _{n)=} (df _{1,...,df} _{n)-\Theta(x)(f} _{1,...,f} ^{t} ^{n)} for any matrix of one-forms \Theta on X. The horizontal sections are then

$$E^\backslash nabla\_U=\; \backslash left\backslash$$ i.e., the solutions to the linear differential equation df _{i} = \sum \Theta _{ij} f _{j} .

If

\Theta extends to a one-form on \Complex the above will also define a local system on \Complex , so will be trivial since \pi _{1(\Complex)} = 0 . So to give an interesting example, choose one with a pole at 0:

in which case for \nabla = d + \Theta , $$E^\backslash nabla\_U\; =\backslash left\backslash$$

• An n-sheeted covering map

X\to Y is a local system with fibers given by the set \{ 1 , ... ,n\} . 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

\pi _{1(X,x)} .

• 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.
• The Gauss–Manin connection is a prominent example of a connection whose horizontal sections are studied in relation to variation of Hodge structures.

Cohomology

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

l{L} of abelian groups on X, we have the sheaf cohomology groups H ^{j(X,l{L})} with coefficients in l{L} .

• Given a locally constant sheaf

l{L} of abelian groups on X, let C ^{n(X;l{L})} be the group of all functions f which map each singular n-simplex \sigma\colon\Delta ^{n\to} X to a global section f(\sigma) of the inverse-image sheaf \sigma ^{-1} l{L} . These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define H ^{j} _{sing (X; l{L})} to be the cohomology of this complex.

• The group

C _{n(\widetilde{X})} of singular n-chains on the universal cover of X has an action of \pi _{1(X,x)} by deck transformations. Explicitly, a deck transformation \gamma\colon\widetilde{X}\to\widetilde{X} takes a singular n-simplex \sigma\colon\Delta ^{n\to\widetilde{X}} to \gamma\circ\sigma . Then, given an abelian group L equipped with an action of \pi _{1(X,x)} , one can form a cochain complex from the groups \operatorname{Hom} _{\pi _{1(X,x)}} (C _{n(\widetilde{X}),L)} of \pi _{1(X,x)} -equivariant homomorphisms as above. Define H ^{j} _{sing (X;L)} to be the cohomology of this complex.

If X is paracompact and locally contractible, then

H ^{j(X,l{L})\congH} ^{j} ^{sing (X; l{L})} .^[3] If l{L} is the local system corresponding to L, then there is an identification C ^{n(X; l{L})\cong\operatorname{Hom}} _{\pi _{1(X,x)}} (C _{n(\widetilde{X}),L)} compatible with the differentials,^[4] so H ^{j} _{sing (X;L)} .

Generalization

Local systems have a mild generalization to constructible sheaves -- a constructible sheaf on a locally path connected topological space

X is a sheaf l{L} such that there exists a stratification ofX = \coprod X _{λ} where l{L}| _{X _{λ}} is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map f:X \to Y . For example, if we look at the complex points of the morphismf:X =Proj \left( (\Complex[s,t][x,y,z])/((stf(x,y,z))) \right) \to Spec (\Complex[s,t]) then the fibers overA ^{2} _{s,t} - V (st) are the smooth plane curve given by f , but the fibers over V are P ^{2} . If we take the derived pushforward R f _{!(\underline{\Q}} _{X)} then we get a constructible sheaf. Over V we have the local systems\begin{align} R ^{0 f} _{!(\underline{ Q} }_X)|_ &= \underline_ \\\mathbf^2f_!(\underline_X)|_ &= \underline_ \\\mathbf^4f_!(\underline_X)|_ &= \underline_ \\\mathbf^kf_!(\underline_X)|_ &= \underline_ \text\endwhile over A ^{2} _{s,t} - V (st) we have the local systems\begin{align} R ^{0 f} _{!(\underline{\Q}} _{X)|} ^{A ^{2} _{s,t} - V (st)} & = \underline{\Q} _{A ^{2} _{s,t} - V (st)} \\ R ^{1 f} _{!(\underline{\Q}} _{X)|} ^{A ^{2} _{s,t} - V (st)} & = \underline{\Q} ^{⊕ 2 g} _{A ^{2} _{s,t} - V (st)} \\ R ^{2 f} _{!(\underline{\Q}} _{X)|} ^{A ^{2} _{s,t} - V (st)} & = \underline{\Q} _{A ^{2} _{s,t} - V (st)} \\ R ^{kf} _{!(\underline{\Q}} _{X)|} ^{A ^{2} _{s,t} - V (st)} & = \underline{ 0 } _{A ^{2} _{s,t} - V (st)} otherwise \end{align} where g is the genus of the plane curve (which is g = (\deg(f) - 1 )(\deg(f) - 2 )/ 2 ).

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.

See also

• Leray spectral sequence
• Gauss–Manin connection
• D-module
• Intersection homology
• Perverse sheaf

External links

• Web site: What local system really is. Stack Exchange.
• Web site: Computing Cohomology of Local Systems. Christian. Schnell. Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.
• Web site: An illustrated guide to perverse sheaves. Geordie. Williamson. Geordie Williamson.
• Web site: Intersection homology and perverse sheaves. Robert . MacPherson . Robert MacPherson (mathematician). December 15, 1990.
• Web site: Local systems and constructible sheaves. Fouad. El Zein. Jawad . Snoussi.

Notes and References

1. Steenrod . Norman E. . Norman Steenrod. Homology with local coefficients . . 44 . 4 . 1943 . 10.2307/1969099 . 610–627 . 9114.
2. [James Milne (mathematician) |Milne, James S.]
3. [Glen Bredon |Bredon, Glen E.]
4. [Allen Hatcher |Hatcher, Allen]