# 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.

## 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 ${\mathcal {L}}$ is a local system if every point has an open neighborhood $U$ such that ${\mathcal {L}}|_{U}$ is a constant sheaf.

### Path-connected spaces

If X is path-connected, a local system ${\mathcal {L}}$ 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

$\rho :\pi _{1}(X,x)\to {\text{Aut}}(L)$ and similarly for local systems of modules,... The map $\pi _{1}(X,x)\to {\text{End}}(L)$ giving the local system ${\mathcal {L}}$ is called the monodromy representation of ${\mathcal {L}}$ .

Proof of equivalence

Take local system ${\mathcal {L}}$ and a loop $\gamma$ at x. It's easy so show that any local system on $[0,1]$ is constant. For instance, $\gamma ^{*}{\mathcal {L}}$ is constant. This gives an isomorphism $(\gamma ^{*}{\mathcal {L}})_{0}\simeq \Gamma ([0,1],{\mathcal {L}})\simeq (\gamma ^{*}{\mathcal {L}})_{1}$ , i.e. between L and itself. Conversely, given a homomorphism $\rho :\pi _{1}(X,x)\to {\text{End}}(L)$ , consider the constant sheaf ${\underline {L}}$ on the universal cover ${\widetilde {X}}$ of X. The deck-transform-invariant sections of ${\underline {L}}$ 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

${\mathcal {L}}(\rho )_{U}\ =\ \left\{{\text{sections }}s\in {\underline {L}}_{\pi ^{-1}(U)}{\text{ with }}\theta \circ s=\rho (\theta )s{\text{ for all }}\theta \in {\text{ Deck}}({\widetilde {X}}/X)=\pi _{1}(X,x)\right\}$ where $\pi :{\widetilde {X}}\to X$ 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

${\mathcal {L}}\colon \Pi _{1}(X)\to {\textbf {Mod}}(R)$ from the fundamental groupoid of $X$ to the category of modules over a commutative ring $R$ . Typically $R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C}$ . What this is saying is that at every point $x\in X$ we should assign a module $M$ with a representations of $\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)$ such that these representations are compatible with change of basepoint $x\to y$ for the fundamental group.

## Examples

• Constant sheaves. For instance, ${\underline {\mathbb {Q} }}_{X}$ . This is a useful tool for computing cohomology since the sheaf cohomology
$H^{k}(X,{\underline {\mathbb {Q} }}_{X})\cong H_{\text{sing}}^{k}(X,\mathbb {Q} )$ is isomorphic to the singular cohomology of $X$ .
• $X=\mathbb {R} ^{2}\setminus \{0,0\}$ . Since $\pi _{1}(\mathbb {R} ^{2}\setminus \{0,0\})=\mathbb {Z}$ , there are $S^{1}$ -many linear systems on X, the $\theta \in \mathbb {R}$ one given by monodromy representation
$\pi _{x}(X)=\mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )$ by sending $n\mapsto e^{in\theta }$ • Horizontal sections of vector bundles with a flat connection. If $E\to X$ is a vector bundle with flat connection $\nabla$ , then
$E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}$ is a local system.
For instance, take $X=\mathbb {C} \setminus 0$ and $E=X\times \mathbb {C} ^{n}$ the trivial bundle. Sections of E are n-tuples of functions on X, so $\nabla _{0}(f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})$ defines a flat connection on E, as does $\nabla (f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})-\Theta (x)(f_{1},\dots ,f_{n})^{t}$ for any matrix of one-forms $\Theta$ on X. The horizontal sections are then
$E_{U}^{\nabla }=\left\{(f_{1},\dots ,f_{n})\in E_{U}:(df_{1},\dots ,df_{n})=\Theta (f_{1},\dots ,f_{n})^{t}\right\}$ i.e., the solutions to the linear differential equation $df_{i}=\sum \Theta _{ij}f_{j}$ .
If $\Theta$ extends to a one-form on $\mathbb {C}$ the above will also define a local system on $\mathbb {C}$ , so will be trivial since $\pi _{1}(\mathbb {C} )=0$ . So to give an interesting example, choose one with a pole at 0:
$\Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}$ in which case for $\nabla =d+\Theta$ ,
$E_{U}^{\nabla }=\left\{f_{1},f_{2}:U\to \mathbb {C} \ \ {\text{ with }}f'_{1}=f_{2}/x\ \ f_{2}'=f_{1}+e^{x}f_{2}\right\}$ • An n-sheeted covering map $X\to Y$ is a local system with sections locally the set $\{1,\dots ,n\}$ . 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 the same as a k-linear representation of the group $\pi _{1}(X,x)$ .
• 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 $X$ is a sheaf ${\mathcal {L}}$ such that there exists a stratification of

$X=\coprod X_{\lambda }$ where ${\mathcal {L}}|_{X_{\lambda }}$ 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 morphism

$f:X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x,y,z]}{(stf(x,y,z))}}\right)\to {\text{Spec}}(\mathbb {C} [s,t])$ then the fibers over

$\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)$ are the smooth plane curve given by $f$ , but the fibers over $\mathbb {V}$ are $\mathbb {P} ^{2}$ . If we take the derived pushforward $\mathbf {R} f_{!}({\underline {\mathbb {Q} }}_{X})$ then we get a constructible sheaf. Over $\mathbb {V}$ we have the local systems

{\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{4}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {0}}_{\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}} while over $\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)$ we have the local systems

{\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{1}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}^{\oplus 2g}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {0}}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}} where $g$ is the genus of the plane curve (which is $g=({\text{deg}}(f)-1)({\text{deg}}(f)-2)/2$ ).

## 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.

### Topological Bivariant Theory

For maps $f\colon X\to Y$ there is a Bivariant theory similar to William Fulton's, called Topological Bivariant Theory. Defining such a theory requires local systems and the six-functor formalism. Bivariant theories are characterized by the property

$H^{i}(X{\xrightarrow {f}}Y)={\text{Hom}}_{D^{b}(Y)}(\mathbb {R} f_{!}\mathbb {Q} _{X},\mathbb {Q} _{Y}[+i])$ For example, this can be computed in some simple cases. If $Y$ is a point, this recovers Borel–Moore homology. If $Y=X$ and the map is the identity, then this is oridinary cohomology. Another informative class of example includes covering spaces. For example, if $f\colon \mathbb {G} _{m}\to \mathbb {G} _{m}=\mathbb {C} ^{*}$ is the degree $d$ covering given by $z\mapsto z^{d}$ . Then, at the stalk level the cohomology groups are of the form

$\mathbb {R} f_{!}(\mathbb {Q} _{\mathbb {G} _{m}})|_{x}\cong \mathbb {Q} ^{\oplus d}$ and the monodromy for $\pi _{1}(\mathbb {G} _{m})=\mathbb {Z}$ is given by the map taking a branch to its next branch and the $d$ -th branch to the first branch. That is, $\phi \colon \mathbb {Z} \to {\text{Aut}}(\mathbb {Q} ^{d})$ is generated by the matrix

$\phi (1)={\begin{pmatrix}0&0&0&\cdots &0&1\\1&0&0&\cdots &0&0\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\0&0&0&\cdots &1&0\end{pmatrix}}$ 